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Similar Shapes MENU Enlargements Similar Triangle Calculations Scale Factors and VOLUME questions What makes shapes similar ? Similar Triangle questions Congruency Match up the Similar rectangles Scale Factors and AREA Congruency questions What makes Triangles similar ? Scale Factors and AREA examples Match up the Similar triangles Scale Factors and AREA questions Similar Shape Calculations Scale Factors and VOLUME Similar Shape questions Scale Factors and VOLUME examples 1 Main menu Enlargements MENU Basic Fractional Enlargements Negative Scale Factor Enlargements Fractional Enlargements questions Negative Scale Factor questions Describing Fractional Enlargements Describing Negative Enlargements Basic questions Positive Whole Scale Factor Enlargements Positive Whole Scale Factor Enlargement questions Describing Positive Whole Scale Factor Enlargements 2 Similar Shapes Menu 3 Menu Twice asyou big.to That’s a Scale Factor = 2 I want enlarge the rectangle. You know exactly what I mean ! How many bigger you want ?! Do you mean O.K. twice Youtimes the mean AREA twice or do the twice line the lengths lineitlengths ? 4 Menu What are the DIMENSIONS of the following enlargements ? S.F. = 2 S.F. = 2 S.F. = 3 5 Menu Copy the following shapes onto squared paper and sketch their enlargements. 2) 1) 3) S.F. = 3 S.F. = 3 4) S.F.= 2 5) S.F.= 4 S.F.= 3 6 Menu 7 Menu I will fix the position of the ‘image’ by using a Centre of All of theis ‘light rays’ from and lead prefer If I wanted enlarge aoriginate shape say a Scale If the enlargement carried out on the a by grid then Enlargement intothe same way that position of you a lensmay fixes the back to the Centre of Enlargement. Factor = 2 I could draw it anywhere ! position an image. to ‘count theofsquares.’ Scale Factor = 2 x x x x C of E 87 4 x =8 x 2x=214 7.284.47 x 2 = x14.56 2 = 8.94 8 Menu If the enlargement is carried out on a grid then you may prefer to ‘count the squares.’ Scale Factor = 2 C of E x x x x 7 4 left7and left 4 left 2 down x x2 2= =148 left left x 2x and =and 2 14 = 844left left down down x 9 Menu x 2.2 x 3 = 6.6 Object 2.2 x Scale Factor = 3 x Image 2.2 x 3 = 6.6 or 2.2 1.4 x 1.4 x 3 = 4.2 x x x 2 11right rightand and12down up x3=3 6 right and 3 6 down up x 10 Menu Scale Factor = 2 x x x x 3.6 x 2 = 7.2 3.6 2.2 3.2 x x 2.21.4x 2 = 4.4 3.2 x 2 = 6.4 1.4 x 2 = 2.8 x x x x 11right right 3 3left left and and and and 121down up 2down up xx 22x==26 22=left right right 6 left and and and and 2 4down 24up down up 11 Menu Enlarge the following shapes by the given Scale Factors. 1) x S.F. = 2 2) 4) X 3) S.F. = 3 X S.F. = 3 x S.F. = 2 12 Menu Answers Fully describe each Enlargement : Give the Scale Factor and the coordinates of the Centre of Enlargement (Each square = 1 unit) x (-9,7) S.F. = 2 1) y 2) (3,4) Object x Object Image Image S.F. = 3 x 3) x (-4,-2) S.F. = 4 Object Image 13 Menu Answers 14 Menu Scale Factor = 1/2 ½ of 16.1 = 8.05 ½ of 10.2 = 5.1 16.1 10.2 C of E x x Object Image x x x 10.2 ½ of 10.2 = 5.1 16.1 10 10 16 left left left left and and and and 222 down 2down up up ½ of 16.1 = 8.05 ½ of 10 16 left and ½ of 2 up = 5 left and 1 up ½ of ½16 of ½ left 10ofleft and 16and left ½ of ½ and 2ofdown ½ 2 down of 2= up 8 =left =5 8left and leftand 1and down 1 1down up Scale factors less than 1 will produce You may prefer to count squares ! images smaller than their objects. x x Object Image x x x 15 Menu Enlarge the following shapes by the given Scale Factors. 1) x 2) S.F.= 1/2 S.F.= 1/4 3) 4) x x x S.F.= 2/3 . S.F.= 1 5 16 Menu Fully describe each Enlargement : Give the Scale Factor and the coordinates of the Centre of Enlargement (Each square = 1 unit) 1) Object 2) S.F. = 1/3 S.F. = 1/2 C of E at ( 3 , 2 ) Image 3) x x C of E at ( - 6 , 0 ) Image C of E at ( - 1 , - 1 ) x Image S.F. = 1/4 Object Object 17 Menu 18 Menu The lens in your eye produces an image using a Negative As with the other enlargements youFactor could! have carried out these negative Scale enlargements by counting squares in the opposite directions. x x Scale Factor = - 2 2.83 x -2 = - 5.66 Image x Object x 2.83 x -2 = - 5.66 4.47 x -2 = - 8.94 19 Menu Enlarge the following shapes by the given Scale Factors. 1) S.F. = - 2 x 2) X S.F. = - 0.5 20 Menu Fully describe each Enlargement : Give the Scale Factor and the coordinates of the Centre of Enlargement (Each square = 1 unit) 1) Enlargement, Scale Factor = - 3 Object Centre of Enlargement at ( - 7 , 4 ) Enlargement, Scale Factor = - 1/4 x 2) Centre of Enlargement at ( 5 , - 1 ) Image Image x Object 21 Menu 22 Menu Two shapes are said to be SIMILAR when one is an ENLARGEMENT of the other. 4 2 1 2 For shapes to be similar they must : 1) Have identical angles. 2) Have their sides in the same proportion. =2 =2 23 Menu Match up the PAIRS of Similar Shapes. 1) 3 2 6÷3=2 10 3) 4) 10 ÷ 2 = 5 6) 2 Rectangles NOT drawn to scale. 6 7 7 ÷ 2 = 3.5 2) 3 2 3 ÷ 2 = 1.5 5) 14 4 7) 3 14 ÷ 4 = 3.5 15 15 ÷ 3 = 5 4 8) 10 8 8÷4=2 15 15 ÷ 10 = 1.5 24 Menu 25 Menu Same angles so automatically Similar Triangles. 0 40 0 40 0 0 50 50 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. 26 Menu Match up the PAIRS of Similar Triangles. Triangles NOT drawn to scale. 1) 2) 3) 40° 80° 30° 60° 70° 4) 20° 70°60° 5) 6) 60° 8) 50° 60° 7) 20° 60° 40° 27 Menu 28 Menu Bob decides to enlarge a poster of himself. How wide will the enlargement be ? 10 cm 15 cm 30 × = x cm × 30 x = 10 × 30 30 cm 15 x = 300 15 x = 20 cm 29 Menu Bob’s work rival decides to reduce the poster so that it is only 3 cm wide. How long will it be ? 20 cm 3× = ×3 3 cm x cm x = 30 × 3 20 30 cm x = 90 20 x = 4.5 cm 30 Menu Calculate the missing lengths. { Each pair of shapes are similar } 1) x 3 2) 15 5 9 9.6 10 8 12 x 3) 8 x 9 16.87 5 17.64 7 5) 15 9 4) 115° 15 5 115° 17 8.75 8 x 6) 6 x 14 5 130° 20 130° 12 x 4 8 31 Menu 32 Menu How high is the church spire ? Common to both triangles Corresponding Angles Parallel Both are Right Angles 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 … 33 Menu How high is the church spire ? 50 × × 50 = x = 2 × 50 4 x = 25 m 2 x 4 2m 4m 50 m 4)6) Measure height of the stick. Let the the height of the spire be called x. 5)7) Measure You may the well distances find it easier from the seeing ‘eye’them to the asstick two separate and the ‘eye’ triangles to the church. 34 Menu Calculate the missing lengths x 1) 5.83 5 x 2) 9.17 17 6 15 11 7 18 11.7 3) 5 x 14 4) 10.8 9 6 x 20 x 18 16 12 35 Menu Harder Problems Calculate the missing lengths x 12.5 1) 2) 6 5 12 x x > 3 1.5 3 4 > 3) 20 A x B 4) Prove that triangles and CDE in ^ ABC ^ (Vertically ACB = DCE opposite angles) ^ = BAC ^ (Alternate 3 CDE angles) question are similar. ^ ^ 26.7 > 10 6 E 5 C 3 > 8 CED = ABC (Alternate angles) D 36 Menu 37 Menu Each rectangle is Areas enlarged using Scale Factor = 2 ? Work In each out the case how of has each theaof Area theincreased rectangles S.F. = 2 2 1 1×2=2 2×3=6 4 5 × 6 = 30 10 7 7 × 10 = 70 2×4=8 4 × 6 = 24 12 6 5 4 6 3 2 2 Diagrams not drawn to scale. 10 10 × 12 = 120 20 14 14 × 20 = 280 ×4 2 8 ×4 6 24 ×4 30 120 ×4 70 280 38 Menu Each is Areas enlarged Factor = 3? Work Inrectangle each out case the how has of using each the aArea ofScale theincreased rectangles S.F. = 3 2 1 1×2=2 2×3=6 6 5 × 6 = 30 10 7 7 × 10 = 70 3 × 6 = 18 6 × 9 = 54 18 6 5 6 9 3 2 3 Diagrams not drawn to scale. 15 15 × 18 = 270 30 21 21 × 30 = 630 ×9 2 18 ×9 6 54 ×9 30 270 ×9 70 630 39 Menu Each is Areas enlarged Factor = 4? Work Inrectangle each out case the how has of using each the aArea ofScale theincreased rectangles S.F. = 4 2 1 1×2=2 2×3=6 8 5 × 6 = 30 10 7 7 × 10 = 70 4 × 8 = 32 8 × 12 = 96 24 6 5 8 12 3 2 4 Diagrams not drawn to scale. 20 20 × 24 = 480 40 28 28 × 40 = 1120 × 16 2 32 × 16 6 96 × 16 30 480 × 16 70 1120 40 Menu ( What is the connection between the 2 Scale Factor Increase in?Area Scale Factor and )the=increase in Area Scale Factor 2 Increase in Area ( Area multiplier ) ×4 3 ×9 4 × 16 41 Menu Example 1 ( Scale Factor )2 = Area multiplier 32 = 9 times Area = ? Area = 8 New Area = 8 × 9 cm2 = 72 cm2 5 cm S.F. = 15 ÷ 5 = 3 15 cm Example 2 ( Scale Factor )2 = Area multiplier Area multiplier = 250 ÷ 10 Area = 10 cm2 4 cm = 25 times Area = 250 cm2 x Scale Factor = Area multiplier S.F. = 25 S.F. = 5 New base : 4 × 5 = 20 cm Menu 42 Work out the following : {All of the shapes are Similar} 1) 96 2) cm2 5 cm2 Area = ? Area Area Area = ? = 45 cm2 = 6 cm2 2 cm 8 cm 4 cm 12 cm 3) 4) Area 8m Area Area = 180 m2 = 100 m2 = 20 m2 Area = 25 m2 15 m x 5m 16 m x 43 Menu 44 Menu Each CUBOID is enlarged using a Scale Factor =2 ? InWork each case how has the Volume increased out the Volume of each cuboid. Diagrams not drawn to scale. S.F. = 2 11 × 1 × 2 1 =2 2 1 1×2×3 2 =6 3 2 2 2 4 4 2×4×6 = 12 4 3 2×3×4 = 24 4 2 16 ×8 6 48 = 48 ×8 4 2×2×3 2 3 = 16 ×8 6 2 2 2×2×4 4 6 4×4×6 12 96 = 96 6 4×6×8 = 192 8 ×8 24 192 45 Menu Each is enlarged using a Scale Factor =3 ? InWork eachCUBOID case how has the Volume increased out the Volume of each cuboid. Diagrams not drawn to scale. S.F. = 3 11 × 1 × 2 1 =2 2 1 1×2×3 2 =6 3 3 3 3 6 6 3×6×9 = 12 6 3 2×3×4 = 24 4 × 27 6 162 = 162 6 2×2×3 2 3 = 54 × 27 2 54 9 2 2 3×3×6 6 9 6×6×9 × 27 12 324 = 324 9 6 × 9 × 12 = 648 12 × 27 24 648 46 Menu Each CUBOID is enlarged using a Scale Factor =4 ? InWork each casethe how has the Volume increased out Volume of each cuboid. Diagrams not drawn to scale. S.F. = 4 11 × 1 × 2 1 =2 2 1 1×2×3 2 =6 3 4 4 4 8 8 4 × 8 × 12 = 12 3 2×3×4 = 24 4 × 64 6 384 = 384 8 2×2×3 2 3 = 128 × 64 2 128 12 2 2 4×4×8 8 8 12 8 × 8 × 12 × 64 12 768 = 768 12 8 × 12 × 16 = 1536 16 × 64 24 1536 47 Menu What is the connection between the Scale 3 = Increase ( Scale Factor ) in Volume Factor and the increase in Volume ? Scale Factor Increase in Volume ( Volume multiplier ) 2 ×8 3 × 27 4 × 64 48 Menu Example 1 5m Volume 10 m ( Scale Factor )3 = Volume multiplier 23 = 8 times Volume =? = 50 m3 New Volume = 50 × 8 = 400 m3 Scale Factor = 2 Example 2 ( Scale Factor )3 = Volume multiplier Volume Volume = 20 m3 = 540 m3 4m Volume multiplier = 540 ÷ 20 = 27 times x 3 Scale Factor = Volume Multiplier New width : 4 × 3 = 12 m 3 S.F. = 27 S.F. = 3 49 Menu Work out the following : {All of the shapes are Similar} 1) 6 120 Volume 2) m3 12 Volume 30 m3 Volume =? = 3750 = 15 m3 20 3) 18 m Volume m3 4) = 40 6 4 9m Volume m3 =? Volume = 1080 m3 Volume x x Volume 36 = 12800 m3 = 200 m3 50 Menu 51 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. 100° 110° 80° 70° 70° 80° 110° 100° 70° 110° 100° 80° 52 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 53 Menu For Triangles to be CONGRUENT ( hence identical ) they have to fulfil one of four conditions : The INCLUDED angle lies between the 2 pairs of equal length sides. 40° 40° 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 54 Menu For Triangles to be CONGRUENT ( hence identical ) they have to fulfil one of four conditions : A corresponding side lies opposite to one of the identical angles. OR 40° 3) 70° 40° 70° 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 55 Menu For Triangles to be CONGRUENT ( hence identical ) they have to fulfil one of four conditions : OR 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 56 Menu Summary of conditions for Congruent Triangles. Side Side Side S.S.S Angle Angle Side A.A.S Side Angle Side S.A.S Right angle Hypotenuse Side R.H.S 57 Menu Which triangles are Congruent to the RED triangle. You must give reasons. ie SSS, AAS, SAS, RHS Triangles not drawn to scale. 1) 9.8 9.8 A.A.S 60° 40° 2) 15 80° 60° 6) 80° 13.2 60° 40° 5) 15 3) 9.8 80° 80° S.S.S 13.2 40° 13.2 15 4) 40° 9.8 15 40° S.A.S 13.2 58 Menu R.H.S problems Which triangles are Congruent to the RED triangle. You must give reasons. ie SSS, AAS, SAS, RHS Triangles not drawn to scale. 37° 5 4 A.A.S 5 37° 53° 3 3 R.H.S 5 3 5 37° 3 53° Menu 5 3 5 53° S.A.S or R.H.S or A.A.S 59 End of Similar Shapes Presentation. Return to previous slide. 60