1 Similar Shapes MENU Main menu Enlargements What makes shapes similar ? Match up the Similar rectangles What makes Triangles similar ? Match up the Similar triangles Similar Shape Calculations Similar Shape questions Similar Triangle Calculations Similar Triangle questions Scale Factors and AREA Scale Factors and AREA examples Scale Factors and AREA questions Scale Factors and VOLUME Scale Factors and VOLUME examples Scale Factors and VOLUME questions Congruency questions

2 Enlargements MENU Similar Shapes Menu Basic Basic questions Positive Whole Scale Factor Enlargements Positive Whole Scale Factor Enlargement questions Describing Positive Whole Scale Factor Enlargements Fractional Enlargements Fractional Enlargements questions Describing Fractional Enlargements Negative Scale Factor Enlargements Negative Scale Factor questions Describing Negative Enlargements

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4 I want you to enlarge the rectangle. How many times bigger do you want it ? Twice as big. That’s a Scale Factor = 2 Do you mean twice the AREA or twice the line lengths ? You know exactly what I mean ! O.K. You mean twice the line lengths !

5 What are the DIMENSIONS of the following enlargements ? S.F. = 2 S.F. = 3 Menu

6 Copy the following shapes onto squared paper and sketch their enlargements. S.F. = 3 S.F.= 4 S.F.= 2 Menu S.F. = 3 1) 2) 3) 4) 5)

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8 If I wanted to enlarge a shape by say a Scale Factor = 2 I could draw it anywhere ! I will fix the position of the ‘image’ by using a Centre of Enlargement in the same way that the position of a lens fixes the position of an image. x C of E 4 Scale Factor = 2 x 2 = 8 x 87 x 2 = 14 x 4.47 x 2 = 8.94 x 7.28 x 2 = x All of the ‘light rays’ originate from and lead back to the Centre of Enlargement. If the enlargement is carried out on a grid then you may prefer to ‘count the squares.’

9 Menu x C of E Scale Factor = 2 xx xx If the enlargement is carried out on a grid then you may prefer to ‘count the squares.’ 4 left x 2 = 8 left 4 left and 2 down x 2 = 8 left and 4 down 7 left x 2 = 14 left 7 left and 2 down x 2 = 14 left and 4 down

10 Menu xx Scale Factor = x 3 = 6.6 x x 3 = 4.2 x x 3 = 6.6 x or 1 right and 2 up x 3 = 3 right and 6 up x 1 right and 1 down x 3 = 3 right and 3 down x 2 right and 1 down x 3 = 6 right and 3 down x ImageObject

11 Menu Scale Factor = 2 xx x 2 = 7.2 x x 2 = 6.4 x x 2 = 4.4 x x 2 = 2.8 x 3 left and 2 up x 2 = 6 left and 4 up x 3 left and 1 down x 2 = 6 left and 2 down x 1 right and 2 up x 2 = 2 right and 4 up x 1 right and 1 down x 2 = 2 right and 2 down x

12 Menu Enlarge the following shapes by the given Scale Factors. x S.F. = 2 1) X S.F. = 3 Answers 2) x S.F. = 2 3) X S.F. = 3 4)

13 Menu Fully describe each Enlargement : Give the Scale Factor and the coordinates of the Centre of Enlargement (Each square = 1 unit) x S.F. = 2 ( - 9, 7 ) y x 1) Object Image Object Image 2) S.F. = 3 x ( 3, 4 ) 3) Object Image S.F. = 4 x ( - 4, - 2 ) Answers

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15 Menu Scale Factor = 1/2 x 10.2 ½ of 10.2 = 5.1 x 16.1 ½ of 16.1 = 8.05 x 10.2 ½ of 10.2 = 5.1 x 16.1 ½ of 16.1 = 8.05 x Scale factors less than 1 will produce images smaller than their objects. C of E Object Image You may prefer to count squares ! x Object 10 left and 2 up ½ of 10 left and ½ of 2 up = 5 left and 1 up x 10 left and 2 down ½ of 10 left and ½ of 2 down = 5 left and 1 down x 16 left and 2 up ½ of 16 left and ½ of 2 up = 8 left and 1 up x 16 left and 2 down ½ of 16 left and ½ of 2 down = 8 left and 1 down x Image

16 Menu Enlarge the following shapes by the given Scale Factors. x S.F.= 1/2 1) x 2) S.F.= 1/4 3) S.F.= 2/3 x S.F.= 1 5. x 4)

17 Menu Fully describe each Enlargement : Give the Scale Factor and the coordinates of the Centre of Enlargement (Each square = 1 unit) Object Image 1) S.F. = 1/2 C of E at ( - 6, 0 ) x x Object Image S.F. = 1/3 C of E at ( 3, 2 ) 2) Object Image S.F. = 1/4 x C of E at ( - 1, - 1 ) 3)

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19 Menu The lens in your eye produces an image using a Negative Scale Factor ! x Scale Factor = x -2 = x 2.83x -2 = x 4.47x -2 = x Object Image As with the other enlargements you could have carried out these negative enlargements by counting squares in the opposite directions.

20 Menu Enlarge the following shapes by the given Scale Factors. x S.F. = - 2 1) X S.F. = )

21 Menu Fully describe each Enlargement : Give the Scale Factor and the coordinates of the Centre of Enlargement (Each square = 1 unit) x Object Image 1) Enlargement, Scale Factor = - 3 Centre of Enlargement at ( - 7, 4 ) x Object Image Enlargement, Scale Factor = - 1/4 2) Centre of Enlargement at ( 5, - 1 )

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For shapes to be similar they must : 1)Have identical angles. 2)Have their sides in the same proportion. 4 2 = Menu Two shapes are said to be SIMILAR when one is an ENLARGEMENT of the other.

24 Match up the PAIRS of Similar Shapes. Rectangles NOT drawn to scale. Menu 1) 2) 3) 4)5) 6) 7) 8) ÷ 3 = 2 8 ÷ 4 = 2 3 ÷ 2 = ÷ 10 = ÷ 2 = 5 15 ÷ 3 = 5 14 ÷ 4 = ÷ 2 = 3.5

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26 For 2 Triangles to be Similar to each other you only need to check whether or not they have the same angles. If their angles are the same then their sides will automatically be in the same proportions Same angles so automatically Similar Triangles. Menu

27 Match up the PAIRS of Similar Triangles. Triangles NOT drawn to scale. Menu 40° 80° 40° 60° 30° 1)2) 3) 4) 5) 6) 7) 8) 60° 70° 20° 60° 70° 60° 50°

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29 10 cm 15 cm Menu x cm 30 cm Bob decides to enlarge a poster of himself. How wide will the enlargement be ? x 30 = × × 30 x = 10 × x = x = 20 cm

30 Menu Bob’s work rival decides to reduce the poster so that it is only 3 cm wide. How long will it be ? x cm 3 cm 20 cm 30 cm x 3 = × × 3 x = 30 × 3 20 x = x = 4.5 cm

31 Calculate the missing lengths. { Each pair of shapes are similar } Menu x x x x 14 x x ° 115° 1) 2) 3) 4) 5) 6)

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33 Menu How high is the church spire ? 1)Hammer a stick into the ground. 2)Line up the top of the stick with the top of the spire. {You will need to put your eye to the ground} 3)We now have 2 Similar Triangles because … Parallel Common to both triangles Both are Right Angles Corresponding Angles

34 Menu How high is the church spire ? 4)Measure the height of the stick. 2 m 5)Measure the distances from the ‘eye’ to the stick and the ‘eye’ to the church. 4 m 50 m 6)Let the height of the spire be called x. x 7)You may well find it easier seeing them as two separate triangles x 50 = ×× 50 x = 2 × 50 4 x = 25 m

35 Menu Calculate the missing lengths x x x x x )2) 3) 4) Harder Problems

36 Menu Calculate the missing lengths x > > > > > > x x x )2) 3) 4) AB C D E Prove that triangles ABC and CDE in question 3 are similar ACB = DCE (Vertically opposite angles) CDE = BAC (Alternate angles) CED = ABC (Alternate angles) ^^ ^^ ^^

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38 Menu Each rectangle is enlarged using a Scale Factor = 2 Diagrams not drawn to scale     Work out the Areas of each of the rectangles S.F. = 2 1 × 2 = 22 × 4 = 8 2 × 3 = 64 × 6 = 24 5 × 6 = × 12 = × 10 = × 20 = 280 In each case how has the Area increased ? 2  8 6    280 × 4

39 Menu Each rectangle is enlarged using a Scale Factor = 3 Diagrams not drawn to scale     Work out the Areas of each of the rectangles S.F. = 3 1 × 2 = 23 × 6 = 18 2 × 3 = 66 × 9 = 54 5 × 6 = × 18 = × 10 = × 30 = 630 In each case how has the Area increased ? 2  18 6    630 × 9

40 Menu Each rectangle is enlarged using a Scale Factor = 4 Diagrams not drawn to scale     Work out the Areas of each of the rectangles S.F. = 4 1 × 2 = 24 × 8 = 32 2 × 3 = 68 × 12 = 96 5 × 6 = × 24 = × 10 = 7028 × 40 = 1120 In each case how has the Area increased ? 2  32 6    1120 × 16

41 What is the connection between the Scale Factor and the increase in Area ? Menu Scale FactorIncrease in Area ( Area multiplier ) 2× 4 3× 9 4× 16 ( Scale Factor ) 2 = Increase in Area

42 Menu Example 1 Area = 8 cm 2 Area = ? 5 cm 15 cm S.F. = 15 ÷ 5 = 3 ( Scale Factor ) 2 = Area multiplier 3 2 = 9 times New Area = 8 × 9 = 72 cm 2 Example 2 ( Scale Factor ) 2 = Area multiplier 4 cmx Area = 10 cm 2 Area = 250 cm 2 Area multiplier = 250 ÷ 10 = 25 times Scale Factor = Area multiplier  S.F. =  25 S.F. = 5 New base : 4 × 5 = 20 cm

43 Menu Work out the following : {All of the shapes are Similar} Area = 6 cm 2 Area = ? 2 cm8 cm Area = ? Area = 45 cm 2 12 cm 4 cm Area = 180 m 2 Area = 20 m 2 x5 m 16 mx Area = 100 m 2 Area = 25 m 2 1)2) 3)4) 96 cm 2 5 cm 2 15 m 8 m

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45 Menu Each CUBOID is enlarged using a Scale Factor = 2 Diagrams not drawn to scale. S.F. = 2     Work out the Volume of each cuboid. 1 × 1 × 2 = 2 2 × 2 × 4 = 16 1 × 2 × 3 = 6 2 × 4 × 6 = 48 2 × 2 × 3 = 12 4 × 4 × 6 = 96 2 × 3 × 4 = 24 4 × 6 × 8 = 192 In each case how has the Volume increased ? 2  16 6    192 × 8

46 Menu Each CUBOID is enlarged using a Scale Factor = 3 Diagrams not drawn to scale. S.F. = 3     Work out the Volume of each cuboid. 1 × 1 × 2 = 2 3 × 3 × 6 = 54 1 × 2 × 3 = 6 3 × 6 × 9 = × 2 × 3 = 12 6 × 6 × 9 = × 3 × 4 = 24 6 × 9 × 12 = 648 In each case how has the Volume increased ? 2  54 6    648 × 27

47 Menu Each CUBOID is enlarged using a Scale Factor = 4 Diagrams not drawn to scale. S.F. = 4     Work out the Volume of each cuboid. 1 × 1 × 2 = 2 4 × 4 × 8 = × 2 × 3 = 6 4 × 8 × 12 = × 2 × 3 = 12 8 × 8 × 12 = × 3 × 4 = 24 8 × 12 × 16 = 1536 In each case how has the Volume increased ? 2     1536 × 64

48 What is the connection between the Scale Factor and the increase in Volume ? Menu Scale Factor Increase in Volume ( Volume multiplier ) 2× 8 3× 27 4× 64 ( Scale Factor ) 3 = Increase in Volume

49 ( Scale Factor ) 3 = Volume multiplier Menu Example 1 Example 2 Volume = 50 m 3 Volume = ? 5 m 10 m Scale Factor = = 8 times New Volume = 50 × 8 = 400 m 3 ( Scale Factor ) 3 = Volume multiplier Volume = 20 m 3 Volume = 540 m 3 4 mx Volume multiplier = 540 ÷ 20 = 27 times Scale Factor =  Volume Multiplier 3 S.F. =  27 3 S.F. = 3 New width : 4 × 3 = 12 m

50 Menu Work out the following : {All of the shapes are Similar} Volume = 15 m Volume = ? Volume = 3750 m 3 Volume = ? 420 Volume = 40 m 3 Volume = 1080 m 3 6 x x 36 Volume = 200 m 3 Volume = m 3 1)2) 3)4) 120 m 3 30 m 3 18 m 9 m

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52 Menu Shapes are said to be CONGRUENT when they have the same angles and their sides are the same length. They are identical. They would fit perfectly over each other. 80° 100° 70° 110° 70°80° 110°100° 110° 80° 70°

53 Menu For Triangles to be CONGRUENT ( hence identical ) they have to fulfil one of four conditions : 1)If their sides are all the same length then the triangles are identical ( Congruent ). Side Side Side S.S.S

54 Menu For Triangles to be CONGRUENT ( hence identical ) they have to fulfil one of four conditions : 2)If 2 of their sides are the same length and their INCLUDED angles are the same then the triangles are identical ( Congruent ). Side Angle Side S.A.S 40° The INCLUDED angle lies between the 2 pairs of equal length sides. 40°

55 Menu For Triangles to be CONGRUENT ( hence identical ) they have to fulfil one of four conditions : 3)If 2 of their angles are the same and also 1 of their corresponding sides are the same then the triangles are identical ( Congruent ). Angle Angle Side A.A.S 40° 70° A corresponding side lies opposite to one of the identical angles. OR

56 For Triangles to be CONGRUENT ( hence identical ) they have to fulfil one of four conditions : 4)If they both have Right angles, they both have the same Hypotenuse and one other side is the same length then the triangles are identical ( Congruent ). Right angle Hypotenuse Side R.H.S Menu OR

57 Menu Summary of conditions for Congruent Triangles. Side Side Side S.S.SSide Angle Side S.A.S Angle Angle Side A.A.S Hypotenuse Right angle Hypotenuse Side R.H.S

58 Menu Which triangles are Congruent to the RED triangle. You must give reasons. ie SSS, AAS, SAS, RHS Triangles not drawn to scale ° 60°40° 60°40° ° ° °40° 9.8 1) 2)3) 4) 5) A.A.S S.S.S 6) 80° 60° 40° S.A.S R.H.S problems

59 Which triangles are Congruent to the RED triangle. You must give reasons. ie SSS, AAS, SAS, RHS Menu Triangles not drawn to scale ° 5 37° A.A.S 5 3 R.H.S 53° ° ° 5 3 S.A.S or R.H.S or A.A.S 37°

60 End of Similar Shapes Presentation. Return to previous slide.