Constructing Perpendicular Lines and Special Angles: A Guide

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Delft University of Technology**We aren't endorsed by this school

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MATH NB4010

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Mathematics

Date

Dec 11, 2024

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Uploaded by CoachElectronAlbatross47

Construction of geometric figures 10.1 Constructing perpendicular lines REVISING PERPENDICULAR LINES In Grade 8, you learnt about perpendicular lines. 1. What does it mean if we say that two lines are perpendicular? 2. Use your protractor to measure the angles between the following pairs of lines. Then state whether they are perpendicular or not. (a) D (b) Y A % )( W C B / LINES THAT FORM WHEN CIRCLES INTERSECT 1. Do the following: (a) Use a compass to draw two overlapping circles of different sizes. (b) Draw aline through the points where the circles intersect (overlap). (¢) Draw aline to join the centres of the circles. (Y & Step (a) Step (b) Step (c)

(d) Use your protractor to measure the angles between the intersecting lines. (e) What can you say about the intersecting lines? 2. Repeat questions 1(a) to (e) with circles that are the same size. 3. What conclusion can you make about a line drawn between the intersection points of two overlapping circles and a line through their centres? USING CIRCLES TO CONSTRUCT PERPENDICULAR LINES Case 1: A perpendicular through a point that is not on the line segment Copy the steps below: You are given line segment MN with point P at a distance from it. You must construct a line that is perpendicular to MN, so that the perpendicular passes through point P. Step 1 Use your compass to draw a circle whose centre is the one end point of the line segment (N) and passes through the point (P). Step 2 Repeat step 1, but make the centre of your circle the other end point of the line segment (M). Step 3 Join the points where the circles intersect: PQ L MN.

Case 2: A perpendicular at a point that is on the line segment Copy the steps below: You are given line segment XY with point Z on it. You must construct a perpendicular Step 1 Use your compass to draw a circle whose Set your compass wider than it was for the circle with centre Z. Draw two circles of the same size whose centres are at the two points where the first (black) circle intersects XY. The two (green) circles will overlap. line passing through Z. centre is Z. Make its radius smaller than x ZX. Note the two points where the circle intersects XY. 7 X Y Step 2 Step 3 Join the intersection points of the two overlapping circles. Mark these points Cand D: CD 1 XY and passes through point Z. PRACTISE USING CIRCLES TO CONSTRUCT PERPENDICULAR LINES In each of the following two cases, copy the line segment, and draw a line that is perpendicular to the segment and passes through point P. 1. A— B P

10.2 Bisecting angles USING CIRCLES TO BISECT ANGLES Work through the following example of using To bisect something means intersecting circles to bisect an angle. Do the “to cut in half”. following steps yourself. You are given ABC. You must bisect the angle. Step 1 Draw a circle with centre B to mark off an equal length on both arms of the angle. Label the points of intersection D and E: DB = BE.

Step 2 Draw two equal circles with centres at D and at E. Make sure the circles overlap. Step 3 Draw a line from B through the points where the two equal circles intersect. This line will bisect the angle. Same construction as in step 3 above PRACTISE BISECTING ANGLES Can you explain why the method above works to bisect an angle? Can you also see that we need not draw full circles, but can merely use parts of circles (arcs) to do the above construction? Copy the following angles and then bisect them without using a protractor: A E . e F

10.3 Constructing special angles without a protractor Angles of 30°, 45°, 60° and 90° are known as special angles. You must be able to construct these angles without using a protractor. CONSTRUCTING A 45° ANGLE You have learnt how to draw a 90° angle and how to Hint: Extend the line to the bisect an angle, without using a protractor. Copy the left of X. line below and use your knowledge on angles and bisecting angles to draw a 45° angle at point X on the line. CONSTRUCTING 60° AND 30° ANGLES 1. What do you know about the sides and angles in an equilateral triangle? 2. Draw two circles with the following properties: e The circles are the same size. D » Each circle passes through the other circle’s centre. * The centres of the circles are labelled A and B. » The points of intersection of the circles are labelled D and E. An example is shown on the right. E 3. Draw in the following line segments: AB, AD and DB. 4. What can you say about the lengths of AB, AD and DB?

S. What kind of triangle is ABD? 6. Therefore, what do you know about A, B and D? 7. Use your knowledge of bisecting angles to create an angle of 30° on the construction you made in question 2. 8. Copy the line segment below and use what you have learnt to construct an angle of 60° at point P on the line segment. /Q CONSTRUCTING THE MULTIPLES OF SPECIAL ANGLES 1. Copy and complete the table below. The first one has been done for you. Angle | Multiples below 360° Angle | Multiples below 360° 30° 30°; 60°% 90°; 120°; 150°; 180°% 45° 210°; 240°; 270°; 300°%; 330° 60° 90° 2. Construct the following angles without using a protractor. You will need to do more than one construction to create each angle. (a) 120° (b) 135° (c) 270° (d) 240° (e) 150°

10.4 Angle bisectors in triangles You learnt how to bisect an angle in Section 10.2. Now you will investigate the angle bisectors in a triangle. An angle bisector is a line that cuts an angle in half. angle bisector 1. (@) Copy the acute triangle below. Bisect each of the angles of the acute triangle. (b) Extend each of the bisectors to the opposite side of the triangle. (¢) What do you notice? 2. (a) Copy the obtuse angle below. Do the same with the obtuse triangle. (b) What do you notice? 3. Compare your triangles with those of two classmates. You should have the same results. You should have found that the three angle bisectors of a triangle intersect at one point. This point is the same distance away from each side of the triangle.

10.5 Interior and exterior angles in triangles WHAT ARE INTERIOR AND EXTERIOR ANGLES? An interior angle is an angle that lies between two sides of a triangle. It is inside the triangle. A triangle has three interior angles. exterior angle interior An exterior angle is an angle between a side of angles a triangle and another side that is extended. It is outside the triangle. Look at APQR. Its three sides are extended to create three exterior angles. Each exterior angle has one interior adjacent angle (next to it) and two interior opposite angles, as described in the following table: Q Exterior angle Interior adjacent angle Interior opposite angles 1 X zandy 2 y xand z 3 z x and y IDENTIFYING EXTERIOR ANGLES AND INTERIOR OPPOSITE ANGLES 1. Copy the following table and name each exterior angle and its two interior opposite angles below. (a) (b) 9/ (0 13 | /\ 10 11/15 14 2 3N 4 6 Ext. ~ Int. opp. £s

2. AABC below has each side extended in both directions to create six exterior angles. (a) Write down the names of the interior angles of the triangle. (b) Since a triangle has three sides that can be extended in both directions, there are two exterior angles at each vertex. Write down the names of all the exterior angles of the triangle. (c) Explain why MBL is not an exterior angle of AABC. (d) Write down two other angles that are neither interior nor exterior. INVESTIGATING THE EXTERIOR AND INTERIOR ANGLES IN A TRIANGLE 1. Consider ALMN. Write down the name of the exterior angle. 2. Use a protractor to measure M the interior angles and the exterior angle. Copy the drawing and write the measurements on the drawing. 3. Useyour findings in question 2 to complete the following sum: N LMN+MLN=... 4. Whatis the relationship between the exterior angle of a triangle and the sum of the interior opposite angles? The exterior angle of a triangle is equal to the sum of the interior opposite angles.

S. Work out the sizes of angles a to f below, without using a protractor. Give reasons for the statements you make as you work out the answers. (@) (b) 45° 23, 40° b / 127° 10.6 Constructing congruent triangles Two triangles are congruent if they have exactly the same shape and size, i.c. they are able to fit exactly on top of each other. This means that all three corresponding sides and three corresponding angles are equal, as shown in the following two pairs: AABC = ADEF and AGHI = AJKL. In each pair, the corresponding sides and angles are equal. MINIMUM CONDITIONS FOR CONGRUENCY To determine if two triangles are congruent, we need a certain number of measurements, but not all of these. Let’s investigate which measurements give us only one possible triangle. 1. Use a ruler, compass and protractor to construct the following triangles. Each time minimum measurements are given. (@) Given three sides: side, side, side (SSS): ADEF with DE =7 cm, DF =6 cm and EF =5 cm. (b) Given three angles: angle, angle, angle (AAA): AABC with A = 80°, B =60° and C = 40°.

(¢) Given one side and two angles side, angle angle (SAA): AGHI with GH =8 cm, G =60°and H =30°. (d) Given two sides and an included angle: side, angle, side (SAS): AJKL with JK =9 cm, K = 130° and KL = 7 cm. (e) Given two sides and an angle that is not included: side, side, angle (SSA): AMNP with MN = 10 cm, M = 50° and PN = 8 cm. () Given aright angle, the hypotenuse and a side (RHS): ATRS with TR 1L RS, RS=7 cm and TS =8 cm. (g) Triangle UVW with UV =6 ccn and VW =4 cm. . Compare your triangles with those of three classmates. Which of your triangles are congruent to theirs? Which are not congruent? . Go back to AMNP (question 1e). Did you find that you can draw two different triangles that both meet the given measurements? One of the triangles will be obtuse and the other acute. Follow the construction steps below to see why this is so. Step 1 Step 2 Construct MN = 10 cm and the 50° angle N is unknown, but NP = 8 cm. Construct at M, even though you do not know the |an arc 8 cm from N. Every point on the length of the unknown side (MP). arc is 8 cm from N. 50° M 10 cm N Step 3 Point P must be 8 cm from N and fall on the unknown side of the triangle. The arc intersects the third side at two points, so P can be at either point. So two triangles are possible, each meeting the conditions given, i.e. MN = 10 cm, NP =8 cm and M = 50°. M 10 cm N

4. Copy and complete the table. Write down whether or not we can construct a congruent triangle when the following conditions are given. Conditions Congruent? Three sides (SSS) Two sides (SS) Three angles (AAA) Two angles and a side (AAS) Two sides and an angle not between the sides (SSA) Two sides and an angle between the sides (SAS) Right-angled with the hypotenuse and a side (RHS) 10.7 Diagonals of quadrilaterals DRAWING DIAGONALS A diagomnal is a straight line inside a figure that joins two vertices of the figure, where the vertices are not next to each other. 1. Look at the quadrilaterals below. The two diagonals of the square have been drawn in: AC and BD. 2. Copy the quadrilaterals below and draw in the diagonals. A D B C Square Rectangle Parallelogram Rhombus Kite Trapezium 3. How many sides does a quadrilateral have? 4. How many angles does a quadrilateral have? 5. How many diagonals does a quadrilateral have?

DIAGONALS OF A RHOMBUS Below are two overlapping circles with centres A and B. The circles are the same size. 1. Construct arhombus inside the circles by joining the centre of each circle with the intersection points of the circles. Join AB. 2. Copy the circles and construct the perpendicular A Pe"’Pe“diC"la" bisectt?r bisector of AB. (Go back to Section 10.1 if you need is a line that cuts another ':”e help.) What do you find? in half at a right angle (90°). 3. Do the diagonals bisect each other? 4. Copy and complete the sentence: The diagonals of a rhombus will always DIAGONALS OF A KITE Below are two overlapping circles with centres D and E. The circles are different sizes. 1. Copy the circles and construct a kite by joining the centre points of the circles to the intersection points of the circles. 2. Draw in the diagonals of the Kite. 3. Mark all lines that are of the same length.

4. Are the diagonals of the kite perpendicular? 5. Do the diagonals of the kite bisect each other? 6. What is the difference between the diagonals of a thombus and those of a kite? DIAGONALS OF PARALLELOGRAMS, RECTANGLES AND SQUARES 1. Draw a parallelogram, rectangle and square onto grid paper. 2. Draw in the diagonals of the quadrilaterals. 3. Indicate on each shape all the lengths in the diagonals that are equal. (Use a ruler.) 4. Use the information you have found to copy and complete the table below. Fill in “yes” or “no”. Quadrilateral Diagonals equal Diagonals bisect Diagonals meet at 90° Parallelogram Rectangle Square 10.8 Angles in polygons USING DIAGONALS TO INVESTIGATE THE SUM OF THE ANGLES IN POLYGONS 1. We can divide a quadrilateral into two triangles by drawing in one diagonal. (a) Copy the polygons below and draw in diagonals to divide each of the polygons into as few triangles as possible. (b) Write down the number of triangles in each polygon. NGRS, Quadrilateral Pentagon Hexagon No. of As 2 Sumof /s 2 x 180° = 360°

Heptagon Octagon Nonagon No. of As Sum of s 2. The sum of the angles of one triangle = 180°. A quadrilateral is made up of two triangles, so the sum of the angles in a quadrilateral = 2 x 180° = 360°. Work out the sum of the interior angles of each of the other polygons above.

1. Match the words in the column on the right with the definitions on the left. Write the letter of the definition next to the matching word. (a) Aquadrilateral that has diagonals that are perpendicular |Kite and bisect each other (b) Aquadrilateral that has diagonals that are perpendicular | Congruent To each other, and only one diagonal bisects the other (c) Aquadrilateral that has equal diagonals that bisect each |Exterior angle other (d) Figures that have exactly the same size and shape Rhombus (e) Divides into two equal parts Ferpendicular (f) Anangle thatis formed outside a closed shape: it is between | Bisect the side of the shape and a side that has been extended (g) Linesthatintersect at 90° Special angles (h) 90°,45°, 50° and 60° Rectangle 2. Copy and complete the sentence: The exterior angle in a triangle is equal to 3. (a) Construct APQR with angles of 30° and 60°. The side between the angles must be & cm. You may use only a ruler and a compass. (b) Will all triangles with the same measurements above be congruent to APQR? Explainh your answer.

Geometry of 2D shapes 11.1 Revision: Classification of triangles 1. Use a protractor to measure the interior angles of each of the following triangles. Write down the sizes of the angles. A I H 2. Classify the triangles in question 1 according to their angle properties. Copy and complete the following statements by choosing from the following types of triangles: acute-angled, obtuse-angled and right-angled. (@) AABCisan triangle, because

(b) AEDFisa triangle, because 3. The marked angles in each triangle below are equal. Copy and complete the following statements and classify the triangles according to angle and side properties. @ A ... .. is an acute isosceles triangle, because and . (b)) A . ... is a right-angled isosceles triangle, because = and_ . © A ... .. is an obtuse isosceles triangle, because =~ and . J M P o K L O N R Q 4. Copy the table below. Say for what kind of triangle each statement is true. If it is true for all triangles, then write “All triangles”. Statement True for: (a) | Two sides of the triangle are equal. (b) | One angle of the triangle is obtuse. (¢) | Two angles of the triangle are equal. (d) | All three angles of the triangle are equal to 60°. (e) | The size of an exterior angle is equal to the sum of the opposite interior angles. (f) | Thelongest side of the triangle is opposite the biggest angle. (g) | The sum of the two shorter sides of the triangle is bigger than the length of the longest side. (h) | The square of the length of one side is equal to the sum of the squares of the other sides. (i) | The square of the length of one side is bigger than the sum of the squares of the other sides. (j) | The sum of the interior angles of the triangle is 180°.

11.2 Finding unknown angles in triangles When you have to determine the size of an unknown angle or length of a shape in geometry, you must give a reason for each statement you make. Complete the example below. In AABC, AC=BCand C = 40°. Find the size of B (shown in the diagram as x). A Statement Reason AC=BC Given ~A=8B 180°=40°+x + x Sum /s A 180° — 40° = 2x SoX= FINDING UNKNOWN LENGTHS AND ANGLES 1. Calculate the sizes of the unknown angles. E I 52° 79° F

3. Calculate the sizes of y and x. 26° 11.3 Quadrilaterals PROPERTIES OF QUADRILATERALS 1. Name the following quadrilaterals. Copy the quadrilaterals and mark equal angles and equal sides in each figure. Use your ruler and protractor to measure angle sizes and lengths where necessary. A B

N o 2P R* Q 2. Copy and complete the following table: is equal. Properties True for the following quadrilaterals 5 s | 2| 9 £ o 2 e | T S s | E| £ | 3 2 > | £ S 5 | = S A = - o 7 = At least one pair of opposite angles yes yes yes yes yes no Both pairs of opposite angles are equal. At least one pair of adjacent angles is equal. All four angles are equal. Any two opposite sides are equal. Two adjacent sides are equal and the other two adjacent sides are also equal. All four sides are equal. At least one pair of opposite sides is parallel. Any two opposite sides are parallel. The two diagonals are perpendicular. At least one diagonal bisects the other one.

The two diagonals bisect each other. The two diagonals are equal. At least one diagonal bisects a pair of opposite angles. Both diagonals bisect a pair of opposite angles. The sum of the interior angles is 360°. 3. Look at the properties of a square and a thombus. (a) Are all the properties of a square also the properties of a rhombus? Explain. (b) Are all the properties of a rhombus also the properties of a square? Explain. (¢) Which statement is true? Write down the statement. A square is a special kind of rhombus. A rhombus is a special kind of square. Look at the properties of rectangles and squares. (a) Are all the properties of a square also the properties of a rectangle? Explain. (b) Are all the properties of a rectangle also the properties of a square? Explain. (c) Which statement is true? Write down the statement. A square is a special kind of rectangle. A rectangle is a special kind of square. Look at the properties of parallelograms and rectangles. (a) Areall the properties of a parallelogram also the properties of a rectangle? Explain. (b) Are all the properties of a rectangle also the properties of a parallelogram? Explain. (¢) Which statement is true? Write down the statement. A rectangle is a special parallelogram. A parallelogram is a special rectangle. Look at the properties of a thombus and a parallelogram. Is a thombus a special kind of parallelogram? Explain. Compare the properties of a kite and a parallelogram. Why is a kite not a special kind of parallelogram? Compare the properties of a trapezium and a parallelogram. Why is a trapezium not a special kind of parallelogram?

UNKNOWN SIDES AND ANGLES IN QUADRILATERALS 1. Determine the sizes of angles a to e in the quadrilaterals below. Give reasons for your answets. 267° N4 @ 6% (e 45’ 2. Calculate the perimeters of the quadrilaterals on the right. Give your answers to two decimal places. 11.4 Congruent triangles DEFINITION AND NOTATION OF CONGRUENT TRIANGLES If two triangles are congruent, then they have exactly the same size and shape. In other words, if you cut out one of the triangles and place it on the other, they will match exactly. If you know that two triangles are congruent, then each side in the one triangle will be equal to each corresponding side in the second triangle. Also, each angle in the one triangle will be equal to each corresponding angle in the second triangle. B

In the triangles on the previous page, you can see that AABC = AXYZ. The order in which you write the letters when stating that two triangles are congruent is very important. The letters of the corresponding vertices between the two triangles must appear in the same position in the notation. For example, the notation for the triangles on the previous page should be: AABC = AXYZ, because it indicates that A= )/Z, B= SA(, C= 2, AB=XY,BC=YZand AC = XZ. It is incorrect to write AABC = AZYX. Although the letters refer to the same triangles, this notation indicates that A = 2, C= )’Z, AB =Z7Y and BC =YX, and these statements are not true. Congruency symbol = means “is congruent to”. Write down the equal angles and sides according to the following notations: 1. AKLM =APQR 2. AFGH = ACST MINIMUM CONDITIONS FOR CONGRUENT TRIANGLES Earlier in this chapter, you investigated the minimum conditions that must be satisfied in order to establish that two triangles are congruent. The conditions for congruency consist of: SSS (all corresponding sides are equal) » SAS (two corresponding sides and the angle between the two sides are equal) e AAS (two corresponding angles and any corresponding side are equal) e RHS (both triangles have a 90° angle and have equal hypotenuses and one other side equal). Decide whether or not the triangles in each pair below are congruent. For each congruent pair, write the notation correctly and give a reason for congruency. 1. ,‘A 2. G J » O

U N Y Q P T | Vv R o | ’ S 6. £ Y 2 D G 3 4 F H C PROVING THAT TRIANGLES ARE CONGRUENT You can use what you know about the minimum conditions for congruency to prove that two triangles are congruent. When giving a proof for congruency, remember the following: » Each statement you make needs a reason. * You must give three statements to prove any two triangles congruent. e Give the reason for congruency. Example: In the sketch on the right: AB || EC and AD = DC. Prove that the triangles are congruent.

Solution: Statement Reason In AABD and ACED: 1) AD=DC Given 2) ADB=CDE Vert. opp. £s 3) BAD=ECD Alt. s (AB|| EC) -.AABD = ACED AAS 1. Copy the table with the sketch, and prove that AACE = ABDE. Statement Reason 2. Copy the table with the sketch, and prove that AWXZ = AYXZ. Y Statement Reason

3. Copy the table with the sketch, and prove that QR = SP. (Hint: First prove that the triangles are congruent.) Statement Reason Q¢ ?R pé ‘s 4. Copy the table with the sketch, and prove that the triangles below are congruent. Then find the size of QM P. Statement Reason 41, 11.5 Similar triangles PROPERTIES OF SIMILAR TRIANGLES ABAC and ADEF below are similar to each other. Similar figures have the same shape, but their sizes can be different. F C

1. (@) Use a protractor to measure the angles in each triangle on the previous page. Then copy and complete the following table: Angle Angle What do you notice? B = D= A= £ = C= F= (b) What can you say about the sizes of the angles in similar triangles? 2. (a) Use aruler to measure the lengths of the sides in each triangle in question 1. Then copy and complete the following table: Length (cm) Length (cm) Ratio BA = DE = BA: DE = =1:1§ BC= DF = BC:DF= = CA= FE = CA:FE= = (b) What can you say about the relationship between the sides in similar triangles? Ratio reminder Youread 2 : 1 as “two to 3. The following notation shows that the triangles are one”. similar: ABAC ||| ADEF. Why do you think we write the first triangle as ABAC and not as AABC? The properties of similar triangles: e The corresponding angles are equal. * The corresponding sides are in proportion. Notation for similar triangles: If AXYZ is similar to APQR, then we write: AXYZ ||| APQR. As for the notation of congruent figures, the order of the letters in the notation of similar triangles indicates which angles and sides are equal. For AXYZ ||| APQR: Angles: X=P,Y=Qand Z=R Sides: XY: PQ=XZ:PR=YZ: QR If the triangles’ vertices were written in a different order, then the statements above would not be true. When proving that triangles are similar, you either need to show that the corresponding angles are equal, or you must show that the sides are in proportion.

WORKING WITH PROPERTIES OF SIMILAR TRIANGLES 1. Decide if the following triangles are similar to each other: (@) (b) A L 30° 126° 10 cm E 16,6 cm 5cm 8,3cm 75° 75 G - M NN 8 cm F 16 cm (© (d) 2. Do the following task: e Usea ruler and protractor to construct the triangles described in (a) to (d) on the next page. » Use your knowledge of similarity to draw the second triangle in each question. » Indicate the sizes of the corresponding sides and angles on the second triangle.

(a) InAFEFG, G =75° EG=4cmand GF=5 cm. AABC is an enlargement of AEFG, with its sides three times longer. (b) In AMNO, M =45°, N =30°and MN = 5 cm. APQR is similar to AMNO. The sides of AMNO to APQR are in proportion 1: 3. (c) ARST s an isosceles triangle. R = 40°, RS is 10 cm and RS =RT. AVWX is similar to ARST. The sides of ARST to AVWX are in proportion 1: — (d) AKLM isright-angled at L, LM is 7 cm and the hypotenuse is 12 cm. AXYZ is similar to AKLM, so that the sides are a third of the length of AKLM. INVESTIGATION: MINIMUM CONDITIONS FOR SIMILARITY Which of the following are minimum conditions for similar triangles? (a) Two angles in one triangle are equal to S two angles in another triangle. (b) Two sides of one triangle are in the X y same proportion as two sides in another triangle. (c) Two sides of one triangle are in the same proportion as two sides in another X triangle, and the angle between the two sides is equal to the angle between the y corresponding sides. 2y (d) Two sides of one triangle are in the same proportion as two sides in another triangle, X and one angle not between the two sides is equal to the corresponding angle in the other triangle. Y 2y

SOLVING PROBLEMS WITH SIMILAR TRIANGLES 1. Line segment QR is parallel to line segment ST. Parallel lines never meet. Two lines are parallel to each other if the distance between them is the same along the whole length of the lines. Copy and complete the following proof that AQRU ||| ATSU: Statement Reason RQT=QTS Alt. /s (QR || ST) QRS = — Vert. opp. 458 -.AQRU ||| ATSU Equal Zs (or AAA) 2. The following intersecting line segments form triangle pairs between parallel lines. (@) Are the triangles in each pair similar? Explain. (b) Write down pairs of similar triangles. (c) Are triangles like these always similar? Explain how you can be sure without measuring every possible triangle pair.

3. Theintersecting lines on the right form triangle pairs between the line segments that are not parallel. Are these triangle pairs similar? Explain why or why not. 4. Consider the triangles below. DE || BC. Copy the table with the sketch, and prove that AABC ||| AADE. Statement Reason A 5. In the diagram on the right, ST isa telephone pole and UV is a vertical stick. The stick is 1 m high and it casts a shadow of 1,7 m (VW). The telephone pole casts a shadow of 5,1 m (TW). Use similar triangles to calculate the height of the telephone pole. 6. How many similar triangles are there in the diagram below? Explain your answer. y C T

11.6 Extension questions 1. AABC on theright is equilateral. D is the B midpoint of AB, E is the midpoint of BC and F is the midpoint of AC. E (a) Prove that ABDE is an equilateral triangle. (b) Find all the congruent triangles. Give a proof for each. D (c) Name as many similar triangles as you can. Explain how you know they are similar. (d) Whatis the proportion of the corresponding F sides of the similar triangles? (e) Prove that DE is parallel to AC. A (f) Is DF parallel to BC? Is EF parallel to BA? Explain. 2. Consider the similar triangles drawn below using concentric circles. Explain why the triangles are similar in each diagram. @) . (b) | H J Vv J L Uwy A

Geometry of straight lines 12.1 Angle relationships Remember that 360° is one full revolution. If you look at something and then turn all the way around so that you are looking at it again, you have turned through an angle of 360°. If you turn only halfway around so that you look at something that was right behind your back, you have turned through an angle of 180°. 1. Answer the questions about the figure below. F (a) Isangle FOD in the figure smaller or bigger than a right angle? (b) Isangle FOE in the above figure smaller or bigger than a right angle? In the figure above, FOD + FOE = half of a revolution = 180°. The sum of the angles on a straight line is 180°. When the sum of angles is 180°, the angles are called supplementary. 2. CMA in the figure on the rightis 75°. p AMB is a straight line. (a) How big is CMB? (b) Why do you say so?

3. PMBin the figure in question 2 is 40°. (a) How bigis CMP? (b) Explain your reasoning. 4. In the figure below, AMB is a straight line and AMC and BMC are equal angles. (a) How big are these angles? (b) How do you know this? C M When one line forms two equal angles where it meets another line, the two lines are said to be perpendicular. Because the two equal angles are angles on a straight line, their sum is 180°, hence each angle is 90°. 5. In the figure below, lines AB and CD intersect at point M. B C M D A (@) Doesitlook as if CMA and BMD are equal? In this chapter, you are (b) Can you explain why they are equal? required to give good reasons for every statement (¢) What does CMA + DMA equal? Why do you say so? you make. (d) Whatis CMA + CMB? Why do you say so?

(e) Isittruethat CMA + DMA = CMA + CMB? () Which angle occurs on both sides of the equation in (e)? 6. Look carefully at your answers to questions 5(c) to (e). Now try to explain your observation in question 5(a). 7. In the figure below, AB and CD intersect at M. Four angles are formed. Angle CMB and angle AMD are called vertically opposite angles. Angle CMA and angle BMD are also vertically opposite. When two straight lines intersect, the vertically opposite angles are equal. A D (a) If angle BMC =125° how big is angle AMD? (b) Why do you say so? LINES AND ANGLES A line that intersects other lines is called a transversal. T T, Lo S, G K L In the above pattern, AB is parallel to CD and EF || GH || KB || LD.

1. Anglesa, b, ¢, d and e are corresponding angles. Do the corresponding angles appear to be equal? 2. Investigate whether or not the corresponding angles are equal by using tracing paper. Trace the angle you want to compare to other angles and place it on top of the other angle to find out if they are equal. What do you notice? 3. Anglesf, h, j, mand n are also corresponding angles. Identify all the other groups of corresponding angles in the pattern. 4. Describe the position of corresponding angles that are formed when a transversal intersects other lines. 5. The following are pairs of alternate angles: gand o0; jand s; and kand r. Do these angles appear to be equal? 6. Investigate whether or not the alternate angles are equal by using tracing paper. Trace the angle you want to compare and place it on top of the other angle to find out if they are equal. What do you notice? 7. ldentify two more pairs of alternate angles. 8. Clearly describe the relative position of alternate angles that are formed when a transversal intersects other lines. 9. Did you notice anything about some of the pairs of corresponding angles when you did the investigation in question 6? Describe your finding. 10.Angles fand o, i and g and k and s are all pairs of co-interior angles. Identify three more pairs of co-interior angles in the pattern. The angles in the same relative position at each 4# && intersection where a straight line crosses two others are called corresponding angles. Angles on different sides of a transversal and % % between two other lines are called alternate angles. Angles on the same side of the transversal and between two other lines are called co-interior & angles. ANGLES FORMED BY PARALLEL LINES Corresponding angles The lines AB and CD shown on the following page never meet. Lines that never meet and are at a fixed distance from one another are called parallel lines. We write AB || CD.

Parallel lines have the same direction, i.e. they form equal corresponding angles with any line that intersects them. C F The line EF cuts AB at G and CD at H. EF is a transversal that cuts parallel lines AB and CD. 1. (@) Look carefully at the angles EGA and EHC in the above figure. They are called corresponding angles. Do they appear to be equal? (b) Measure the two angles to check if they are equal. What do you notice? 2. Suppose EGA and EHC are really equal. Would EGB and EHD then also be equal? Give reasons to support your answer. When two parallel lines are cut by a transversal, the corresponding angles are equal. Alternate angles The angles BGF and CHE below are called alternate angles. They are on opposite sides of the transversal. B

3. Do you think angles AGF and DHE should also be called alternate angles? 4. Do you think alternate angles are equal? Investigate by using the tracing paper like you did previously, or measure the angles accurately with your protractor. What do you notice? When parallel lines are cut by a transversal, the alternate angles are equal. 5. Try to explain why alternate angles are equal when the lines that are cut by a transversal are parallel, keeping in mind that corresponding angles are equal. By answering the following questions, you should be able to see how you can explain why alternate angles are equal when parallel lines are cut by a transversal. 6. Are angles BGH and DHF in the figure corresponding angles? What do you know about corresponding angles? E C F 7. (a) What can you say about BGH + AGH? Give a reason. (b) What can you say about DHG + CHG? Give a reason. (¢) Isittrue that BGH+ AGH = DHG + CHG? Explain. (d) Will the equation Ain (c) still be true if you replace angle BGH on the left-hand side with angle CHG? 8. Look carefully at your work in question 7 and write an explanation why alternate angles are equal, when two parallel lines are cut by a transversal. Co-interior angles The prefix “co-” means The angles AGH and CHG in the figure on the following together. The word . . “co-interior” means on the page are called co-interior angles. They are on the same same side side of the transversal. '

C F 9. (a) What do you know about CHG + DHG? Explain. (b) What do you know about BGH + AGH? Explain. (¢) What do you know about BGH + CHG? Explain. (d) What conclusion can you draw about AGH + CHG? Give detailed reasons for your conclusion. When two parallel lines are cut by a transversal, the sum of two co-interior angles is 180°. Another way of saying this is to say that the two co-interior angles are supplementary. 12.2 Identify and name angles 1. In the figure below, the line RF is perpendicular to AB. R B C F (a) IsRF also perpendicular to CD? Justify your answer. (b) Name four pairs of supplementary angles in the figure. In each case, say how you know that the angles are supplementary.

(¢) Name four pairs of co-interior angles in the figure. (d) Name four pairs of corresponding angles in the figure. (e) Name four pairs of alternate angles in the figure. 2. Now you are given that AB and CD in the figure below are parallel. C F (a) Ifitisalso given that RF is perpendicular to AB, will RF also be perpendicular to CD? Justify your answer. (b) Name all pairs of supplementary angles in the figure. In each case, say how you know that the angles are supplementary. (¢) Suppose EGA = x. Give the size of as many angles in the figure as you can, in terms of x. Each time give a reason for your answer. 12.3 SOIVing prOblemS When you solve problems in geometry you can use a short- 1. Line segments AB and CD in the figure below hand way to write your reasons. are parallel. EF and IJ are also parallel. Copy the For example, if two angles are equal because they are corresponding angles, then you can write (corr Zs, AB || CD) as the reason. figure and mark these facts on the figure, and then answer the questions.

(a) Name five angles in the figure that are equal to GHD. Give a reason for each of your answers. (b) Name all the angles in the figure that are equal to AGH. Give a reason for each of your answeirs. 2. ABand CD in the figure below are parallel. EF and IJ are also parallel. NMB = 80° and JLF =40°. C F Find the sizes of as many angles in the figure as you can, giving reasons. 3. In the figure below, AB || CD; EF || AB; JR || GH. You are also given that PMN = 60°, RND = 50°. L F J G S 123 2 3 B 1 2 > 3P 2 E H > 1 D 4 ,N A AN M 1 7 6|5 R ¢ H (a) Find the sizes of as many angles in the figure as you can, giving reasons. (b) Are EF and CD parallel? Give reasons for your answers.

Pythagoras’ Theorem 13.1 Investigating the sides of a right-angled triangle A theorem is a rule or a statement that has been proved through reasoning. Pythagoras’ Theorem is a rule that applies only to right-angled triangles. The theorem is named after the Greek mathematician, Pythagoras. A right-angled triangle has one 90° angle. The longest side of the right-angled triangle is called the hypotenuse. hypotenuse The hypotenuse is the side opposite the 90° angle in a E right-angled triangle. It is always the longest side. hypotenuse /Q\ D G H Pythagoras (569-475 BC) Pythagoras was an influential mathematician. Like many Greek mathematicians of 2 500 years ago, he was also a philosopher and a scientist. He formulated the best-known theorem, today known as Pythagoras’ Theorem. However, the theorem had already been in use 1 000 years earlier, by the Chinese and the Babylonians. J hypotenuse How to say it: “high - pot - eh - news” INVESTIGATING SQUARES ON THE SIDES OF RIGHT-ANGLED TRIANGLES 1. The figure shows a right-angled triangle with squares on each of the sides. (a) (b) (c) Write down the areas of the following: Square A Square B Square C Add the area of square B and the area of square C. What do vou notice about the areas?

2. The figure below is similar to the one in question 1. The lengths of the sides of the right-angled triangle are 5 cm and 12 cm. (a) Whatis the length of the hypotenuse? Count the squares. (b) Use the squares to find the following: Area of A Area of B Area of C Area of B + Area of C (¢) What do you notice about the areas? Is it similar to your answer in 1(c)? N o 3) 3. Aright-angled triangle has side lengths of 8 cm and 15 cm. Use your findings in the previous questions to answer the following questions: (a) Whatis the area of the square drawn along the hypotenuse? (b) What is the length of the triangle’s 15cm hypotenuse? In the previous activity, you should have discovered 8 cm Pythagoras’ Theorem for right-angled triangles. Pythagoras’ Theorem says: In aright-angled triangle, a square formed on the hypotenuse will have the same area as the sum of the side 2 area of the two squares formed on the other sides of hypotenuse the triangle. Therefore: side 1 (Hypotenuse)? = (Side 1)% + (Side 2)? 13.2 Checking for right-angled triangles Pythagoras’ Theorem applies in two ways: » Ifa triangle is right-angled, the sides will have the following relationship: (Hypotenuse)?= (Side 1)% + (Side 2)2. o If the sides have the relationship: (Longest side)?= (Side 1) + (Side 2)?, then the triangle is a right-angled triangle. S0, we can test if any triangle is right-angled without using a protractor.

Example: Is a triangle with sides 12 cm, 16 cm and 20 cm right-angled? 16 cm 12 cm 20 cm (Longest side)? = 20 = 400 cm? (Side 1) + (Side 2)2=12%2 + 162> = 144 + 256 = 400 cm? (Longest side)?= (Side 1)% + (Side 2)? . The triangle is right-angled. ARE THESE RIGHT-ANGLED TRIANGLES? 1. This triangle’s side lengths are 29 mm, 20 mm and 21 mm. (a) Prove thatitis a right-angled triangle. (b) Copy the triangle and mark the right angle in the diagram. 2. Use Pythagoras’ Theorem to determine whether these triangles are right-angled. All values are in the same units. (a) (b) (©) 4 12 3 3. Determine whether the following side lengths would form right-angled triangles. All values are in the same units. (@ 7,9and 12 (b) 7,12 and 14 (c) 16,8and 10 (d) 6,8and 10 (e) 8,15and17 (f) 16,21 and 25

13.3 Finding missing sides You can use the Pythagoras’ Theorem to find the lengths of missing sides if you know that a triangle is right-angled. FINDING THE MISSING HYPOTENUSE Example: Calculate the length of the hypotenuse if the lengths of the other two sides are six units and eight units. AABC is right-angled, so: A AC?> = AB%+B(C? , = (62 + 82) units? 6 ' = 36 + 64 units? = 100 units? [ 1 _ B 3 C AC = 100 units = 10 units Sometimes the square root of a number is not a Surd form whole number or a simple fraction. In these cases, You pronounce surd so that it you can leave the answer under the square root rhymes with word. sign. This form of the number is called a surd. V5 is an example of a number in surd form. \/9 is not a surd because Example: Calculate the length of the hypotenuse you can simplify it: of AABC if B = 90°, AB = two units and BC = five units. J9=3 Leave your answer in surd form, where applicable. Remember when taking the square root that length is always positive. A AC? = AB? + BC? ) ? = 22 + 52 units? = 4 + 25 units? ] = 29 units? B 5 C AC = /29 units 1. Find the length of the hypotenuse in each of the triangles shown on the following page. Leave the answers in surd form where applicable.

() K (b) P 48 m 56 m O 8 cm |1 N L 6cm M © s (d) X Zunits Y | Tm 12 units — T Tm R Z . A rectangle has sides with lengths of 36 mm and 77 mm. Find the length of the rectangle’s diagonal. 36 mMmm 77 mm . AABChas A =90°, AB=3 cm and AC = 5 cm. Make a rough sketch of the triangle, and then calculate the length of BC. . Arectangular prism is made of glass. It has a length of 16 cm, a height of 10 cm and a breadth of 8 cm. ABCD and EFGH are two of its faces. AACH has been drawn inside the prism. Is AACH right-angled? Answer the questions to find out. H 16 cm G D C 10 cm ¥ : 8 cm

(a) Calculate the length of the sides of AACH. Note that all three sides of the triangles are diagonals of rectangles. AC is in rectangle ABCD, AH is in ADHE and HC is in HDCG. (b) Is AACH right-angled? Explain your answer. FINDING ANY MISSING SIDE IN A RIGHT-ANGLED TRIANGLE Example: Find the length of TS in the triangle below. T US? = TU? + TS? 10% = 824+ TS? 8 cm 100 = 64 + TS? 36 = TS? U 10 cm > \/g =TS TS = 6cm 1. Intheright-angled triangles below, calculate the length of the sides that have not been given. Leave your answers in surd form where applicable. (@) b s (© }_‘ 25 [] z 24 X 24 20 Y 26 2. Calculate the length of the third side of each of the following right-angled triangles. First draw a rough sketch of each of the triangles before you do any calculations. Round off your answers to two decimal places. (@) AABChas AB=12cm, BC=18 cm and A =90°. Calculate AC. (b) ADEF has F =90°, DE = 58 cm and DF = 41 cm. Calculate EF. (c) AJKL has K= 90°, JK =119 m and KL = 167 m. Calculate JL. (d) APQRhasPQ=2cm, QR=8 cm and Q =90°. Calculate PR. 3. (a) Aladder with alength of 5 m is placed at an angle against a wall. The bottom of the ladder is 1 m away from the wall. How far up the wall will the ladder reach? Round off to two decimal places. (b) If theladder reaches a height of 4,5 m against the wall, how far away from the wall was it placed? Round off to two decimal places.

PYTHAGOREAN TRIPLES Sets of whole numbers that can be used as the sides of a right-angled triangle are known as Pythagorean triples, for example: 3-4-5 5-12-13 7-24-25 16-30-34 20-21-29 You extend these triples by finding multiples of them. For examples, triples from the 3-4-5 set include the following: 3-4-5 6-8-10 9-12-15 12-16-20 There are many old writings that record Pythagorean triples. For example, from 1900 to 1600 BC, the Babylonians had already calculated very large Pythagorean triples, such as: 1 679-2 400-2 929. How many Pythagorean triples can you find? What is the largest one you can find that is not a multiple of another one? 13.4 More practice using Pythagoras’ Theorem 1. Four lines have been drawn on the grid below. Each square is one unit long. Calculate the lengths of the lines: AB, CD, EF and GH. Do the calculations and write the answers in surd form. ( E / \\\ \\\ H A F K 6 cm N 2. (a) Calculate the area of rectangle KLMN. (b) Calculate the perimeter of AKLM. 4,5 cm L M 3. ABCDisarectangle with AB=4 cm, BC=7 cm and CQ = 1,5 cm. Round off your answers to two decimal places if the answers are not whole numbers.

4 cm 1,5cm B 7cm (a) Whatis thelength of QD? (b) If CP=4,2 cm, calculate the length of PQ. (¢) Calculate the length of AQ and the area of AAQD. 4. MNST is a parallelogram. NR =9 mm and MR = 12 mm. (a) Calculate the area of AMNR. (b) Calculate the perimeter of MNST. 12 mm - N 9mm R PYTHAGORAS’' THEOREM AND OTHER TYPES OF TRIANGLES 30 mm S Pythagoras’ Theorem works only for right-angled triangles. But we can also use it to find out whether other triangles are acute or obtuse. e [f the square of the longest side is /ess than the sum of the squares of the two shorter sides, the biggest angle is acute. For example, in a 6-8-9 triangle: 62 + 82 =100 and 92 = 81. 81 is less than 100 .. the 6-8-9 triangle is acute. e If the square of the longest side is more than the sum of the squares of the two shorter sides, the biggest angle is obtuse. For example, in a 6-8-11 triangle: 6 + 82 =100 and 112 =121. 121 is more than 100 .. the 6-8-11 triangle is obtuse. c Copy and complete the following table. It is based on the triangle on the right. ‘ Decide whether each triangle described is right-angled, acute or obtuse. b a | b | c a’ + b? c? :L";: . Type of triangle 35| 6 |32+52=9+25=34 6>=36 |a’+Db*<c? Acute 2 | 4| 6 a?+b.....c? S 1712 a*+b.....2 12 5 |13 a+b,....c? 12 1 16 | 20 | 122+ 162=144 + 256 =400 | 20> =400 | a? + b? = 2 Right-angled 719 |1 az+b.....c? 8 [ 12|13 a+b,....c?

. Write down Pythagoras’ Theorem in the way that you best understand it. . Calculate the lengths of the missing sides in the following triangles. Leave the answers in surd form if necessary. (a) D 4 cm 6 cm . ABCD is a parallelogram. (a) Calculate the perimeter of ABCD. (b) Calculate the area of ABCD. (b) P 13 m 12 m (1R A [ ] 15 m 12 m -

Area and perimeter of 2D shapes 14.1 Area and perimeter of squares and rectangles REVISING CONCEPTS 1. Each block in figures A to F below measures 1 cm x 1 cm. What is the perimeter and area The perimeter (P) of a shape is the distance along the sides of the shape. of each of the figures? The area (A) of a figure is the size of Copy and complete the table below. the flat surface enclosed by the figure. A C D F G H 2cm 2cm 2cm . . Numberof 1Tcm x 1 cm Figure Perimeter Area squares A g|IO|w

Figure Perimeter Area Numberof Tcm x 1 cm squares E F G H 2. Consider the rectangle below on the right-hand side. It is formed by tessellating identical squares that are 1 cm by 1 cm each. The white part has squares that are hidden. (a) Write down, without counting, the total number of squares that form this rectangle, including those that are hidden. Explain your reasoning. (b) Whatis the area of the rectangle, including the white part? Area of arectangle = length x breadth =Ixb Area of a square = Ix [ = P To tessellate means to cover a surface with identical shapes in such a way that there are no gaps or overlaps. Another word for tessellating is tiling. Both length (/) and breadth (b) are expressed in the same unit. 3. Sipho and Theunis each paint a wall to earn some money during the school holidays. Sipho paints a wall 4 m high and 10 m long. Theunis’s wall is 5§ m high and 8 m long. Who should be paid more? Explain. 4. Whatis the area of a square with a length of 12 mm? 5. The area of a rectangle is 72 cm? and its length is 8 cm. What is its breadth? 14.2 Area and perimeter of composite figures BREAKING UP FIGURES AND PUTTING THEM BACK TOGETHER AGAIN 1. The diagram on the left on the following page shows the floor plan of a room. We can calculate the area of the room by dividing the floor into two rectangles, as shown in the diagram on the right on the following page.

22 m 14 m 9m 9m 15 m 15m 8m 8m Area of the room Area of yellow rectangle + Area of red rectangle = (Ixb)+(IxDb) = (14x9)+(15x8) = 126+ 120 = 246 m? (a) The yellow part of the room has a wooden floor and the red part is carpeted. What is the area of the wooden floor? What is the area of the carpeted floor? (b) Calculate the area of the room dividing the floor into two other shapes. Draw a sketch. 2. Calculate the area of the figures below. 1 cm 5cm |_ J 0,5cm 4 cm A B 3,5cm 1 cm 3 cm 3cm 3. Which of the following rules can be used to calculate the perimeter (P) of a rectangle? Explain. * Perimeter=2x (I+b) l and b refer to the length e Perimeter=I+b+1+b and the breadth of a e Perimeter =21+ 2b rectangle. e Perimeter=I1+b The following are equivalent expressions for perimeter: P=2l+2bandP=2(I+b)and P=1+b+1+Db 4. Check with two classmates that the rule or rules you have chosen above are correct; then apply it to calculate the perimeter of figure A. Think carefully! 5. The perimeter of a rectangle is 28 cm and its breadth is 6 cm. What is its length?

14.3 Area and perimeter of circles REVISING CONCEPTS FROM PREVIOUS GRADES The perimeter of a circle is called the circumference of a circle. You will remember the following about circles from previous grades: » The distance across the circle through its centre is called the diameter (d) of the circle. centre ~ » The distance from the centre of the circle to any point on the circumference is called the radius (7). » The circumference (¢) of a circle divided by its diameter is equal to the irrational value we call pi (T). To simplify calculations, we often use the approximate values: n=3,14 or 2—72 . circumference (¢) The following are important formulae to remember: e d=2randr= %d e Circumference of a circle (c) = 27r e Area of a circle (A) = Tir? CIRCLE CALCULATIONS In the following calculations, use T = 3,14 and round off your answers to two decimal places. If you take a square root, remember that length is always positive. 1. Calculate the perimeter and area of the following circles: (@) Acircle with a radius of 5 m (b) A circle with a diameter of 18 mm 2. Calculate the radius of a circle with: (@) acircumference of 53 cm (b) a circumference of 206 mm 3. Work out the area of the following shapes: A B d=4=3cm

4, Calculate the radius and diameter of a circle with: (a) an area of 200 m? (b) an area of 1 000 m? S. Calculate the area of the shaded part. 14.4 Converting between units CONVERTING BETWEEN UNITS USED FOR PERIMETER AND AREA Always make sure that you use the correct units Remember: in your calculations. Practise the conversions Tem=10mm 1mm=0,1 cm below. Tm=100cm 1cm=0,01m Tkm=17000m 1m=0,001 km 1. Copy and complete the following conversions: (@ 34cm=. ... mm () 501m=,........km (© 226m= ... cm @ 0,58km=_ ... m @ L9em= ... mm ® 73mm=_ .. cm ® 924mm= ... m (h) 32,23km= ..., .. m Remember, to convert between square units, you can use method shown below: To convert cm? to m?: Example Convert 50 cm? to m? lcm? = 1cmx1cm 1cm? = 0,0001 m? = 0,01 m x 0,01 m ~. 50 cm? = 50 x 0,0001 m? = 00,0001 m? = 0,005 m? 2. Convert to cm?; (@) 650 mm? (b) 1200 mm? (c) 18 m? (d) 0,045 m? (e) 93 mm? () 177 m? 3. (a) Convert 93 mm? to m2. (b) Convert 0,017 km? to m?.

14.5 Area of other quadrilaterals PARALLELOGRAMS As shown below, a parallelogram can be made into a rectangle if a right-angled triangle from one side is cut off and moved to its other side. base base W/ base So we can find the area of a parallelogram using the formula for the area of a rectangle: Area of rectangle = Ix b = (base of parallelogram) x (perpendicular height of parallelogram) Area of parallelogram = Area of rectangle . Area of parallelogram = base x perp. height We can use any side of the parallelogram as the base, but we must use the 1. (@) Copy the parallelogram above. perpendicular height on the (b) Using the shorter side as the base of the side we have chosen. parallelogram, follow the steps above to derive the formula for the area of a parallelogram. 2. Work out the area of the following parallelograms using the formula: A B C 10 cm / cm 4 G / / 12 cm % cm 200 mm 15 cm 15 cm 3. Work out the area of the parallelograms. Use the Pythagoras’ Theorem to calculate the unknown sides you need. Remember to use the pre-rounded value for height and then round the final answer to two decimal places where necessary. (o) 0 3 -

/ 5cm |_I 3cm 8 cm 2cm 5¢cm C 12 m D 8 cm 9m 15cm Q T 10 cm RHOMBI A rhombus is a parallelogram with all its sides equal. In the same way we derived the formula for the area of a parallelogram, we can show the following: Area of arhombus = length x perp. height / 7 1. Show how to derive the formula for the area of a rhombus. 2. Calculate the areas of the following rhombi. Round off answers to two decimal places where necessary. A B / 15 cm 7 cm o 3 10 m ----- \ 9 cm 2 cm

KITES To calculate the area of a kite, you use one of its properties, namely that the diagonals of a kite are perpendicular. D Area of kite DEFG = Area of ADEG + Area of AEFG 1 1 = E(bxh)+ E(bxh) 1 1 = 5 (EGxOD) + 3 (EG x OF) 1 =3 EG(OD + OF) 1 =3 EG x DF Notice that EG and DF are the diagonals of the kite. 1 ~ Area of a kite = 7 (diagonal, x diagonal,) 1. Calculate the area of kites with the following diagonals. Give your answers in m?2. (@) 150 mm and 200 mm (b) 25 cm and 40 cm 2. Calculate the area of the kite. A BO=0D =6 cm OC=15cm AD =10 cm - B o) D C TRAPEZIUMS A trapezium has two parallel sides. If we tessellate (tile) two trapeziums, as shown in the diagram on the following page, we form a parallelogram. (The yellow trapezium is the same size as the blue one. The base of the parallelogram is equal to the sum of the parallel sides of the trapezium.)

side 1 side 1 side 2 side 2 side 2 side 1 We can use the formula for the area of a parallelogram to work out the formula for the area of a trapezium as follows: base x height (side 1 + side 2) x height Area of parallelogram 1 Area of trapezium = 7 area of parallelogram 1 D) (side 1 + side 2) x height 1 ~ Area of a trapezium = 7 (sum of parallel sides) x perp. height Calculate the area of the following trapeziums: 20 mm 65 cm A & 0 2 B . E19mm 22 mm E52cm I10 2 mm ) 105 cm ” AREAS OF COMPOSITE SHAPES Calculate the areas of the following 2D shapes. Round off your answers to two decimal 10 m places where necessary. () (b) 12m A 1}2\ cm D Q el Y >cm 4 cm |6 m B > = i E ¢ R > 18 T K P F 20 cm 13 cm : /’\\ 1o/m/; // On — H X X\/\m J \\ S \m = o ——N | G 6 m! //// PO =5cm M

14.6 Doubling dimensions of a 2D shape Remember that a 2D shape has two dimensions, namely length and breadth. You have used length and breadth in different forms, to work out the perimeters and areas of shapes, for example: » length and breadth for rectangles and squares e bases and perpendicular heights for triangles, thombi and parallelograms » two diagonals for Kites. But how does doubling one or both of the dimensions of “Doubling” means to a figure affect the figure’s perimeter and area? multiply by 2. The four sets of figures on the next page are drawn on a grid of squares. Each row shows an original figure; the figure with one of its dimensions doubled, and the figure with both of its dimensions doubled. Each square has a side of one unit. 1. Work out the perimeter and area of each shape. Round off your answers to two decimal places where necessary. 2. Which figure in each set is congruent to the original figure? 3. Copy the table below and fill in the perimeter (P) and area (A) of each figure: Figure Original figure Figure with both dimensions doubled P= P: A A= A: P= P: B A= A: P= P= C A= A: P= P: D A= A= 4. Look at the completed table. What patterns do you notice? Choose one: When both dimensions of a shape are doubled, its perimeter is doubled and its area is doubled. When both dimensions of a shape are doubled, its perimeter is doubled and its area is four times bigger.

. . . One dimension Both dimensions QuliEeEl e doubled doubled 3 6 6 1 1 " P = P = D A= A = § A - \\\ \\ \\\ o~ y4 \ 4 U |0/ P = \\ \\ A = N 4 8 P = P = A = A = 4 8 8 /13 / /13 / / / u u 6 P = V= . h u A = A= P = Diagonal 1 = 4 Diagonal 1 =(8 Diagonal 1 = 8 Diagonal 2 = € Diagonal 2 =6 Diagonal 2 =12 \ / P = \/ P = A = P= A=

1. Write down the formulae for the following: Perimeter of a square Perimeter of a rectangle Area of a square Area of a rectangle Area of a triangle Area of a rhombus Area of a kite Area of a parallelogram Area of a trapezium Diameter of a circle Circumference of a circle Area of a circle 2. (a) Calculate the perimeter of the square and the area of the shaded parts of the square: (b) Calculate the area of the kite: J 5m /7 m <> M