imilar shapes 33 HTR 33.1 imilar triangles Triangle and triangle have the same shape but not the same size. They are called similar triangles. The angles in triangle are the same as the angles in triangle, and so two similar triangles have equal angles. and are a pair of corresponding sides. and and and are also pairs of corresponding sides. In general, if two shapes are similar, the lengths of pairs of corresponding sides are in the same proportion. In this case, that means xample 1 Triangles and F are similar. a Find the value of x. b Work out the length of i ii. olution 1 a x 34 b i F 34 10 cm 12 cm imilar triangles have equal angles. F 21 cm The lengths of pairs of corresponding sides are in the same proportion. x 1 2 10 8 10 12 1 8 F ii F 21 1 2 8 8 21 1 12 ubstitute the known lengths. Rearrange the equation to work out the length of. The lengths of pairs of corresponding sides are in the same proportion. ubstitute the known lengths. Rearrange the equation to work out the length of. 531
HTR 33 imilar shapes In the diagram, and are straight lines. The line is parallel to the line. ngle angle. ngle angle. They are both pairs of corresponding angles. lso, angle is common to both triangle and triangle. o triangle and triangle have equal angles and are similar triangles. The lengths of their corresponding sides are in the same proportion, that is 532 xample 2 R is parallel to T. R and T are straight lines., R, T, T 4.. a alculate the length of R. b alculate the length of T. olution 2 a R R T 8 5 R 6 R 6 8 9. 5 b Method 1 R T 8 5 4. 5 8 4.5 7.2 cm 5 T 7.2 cm 4. T 2.7 cm 4. T R The lengths of pairs of corresponding sides are in the same proportion. ubstitute the known lengths. Rearrange the equation to work out the length of R. The lengths of pairs of corresponding sides are in the same proportion. ubstitute the known lengths. Rearrange the equation to work out the length of. To find the length of T, subtract the length of T from the length of.
33.1 imilar triangles HTR 33 Method 2 R T 8 5 x 4.5 4.5 8 4.5 5(x 4.5) 36 5x 22.5 5x 13.5 x 2.7 T 2.7 cm Let represent the length of T so (x 4.5) cm will represent the length of. ubstitute the known lengths and x 4.5 for. olve the equation for x. tate the length of T. In the diagram, and are straight lines. The line is parallel to the line. Now, angle angle angle angle as they are both pairs of alternate angles. lso, angle angle because, where two straight lines cross, the opposite angles are equal. o triangle and triangle have equal angles and are similar triangles. The lengths of their corresponding sides are in the same proportion, that is xample 3 is parallel to. and are straight lines. 10 cm, 7 cm,,. a alculate the length of. b alculate the length of. 7 cm olution 3 a 1 0 6 3 3 10 5cm 6 10 cm The lengths of pairs of corresponding sides are in the same proportion. ubstitute the known lengths. Rearrange the equation to work out the length of. Notice that in triangle, 1 2 and so in triangle, 1 2. 533
HTR 33 b 1 0 6 7 6 7 4.2 cm 10 imilar shapes The lengths of pairs of corresponding sides are in the same proportion. ubstitute the known lengths. Rearrange the equation to work out the length of. xercise 33 1 Triangles and F are similar. a Find the size of angle F. b Work out the length of i F ii. 2 cm 36 3. 7. F 2 Triangles R and TU are similar. alculate the length of a U b R. 3. R T 3.9 cm U 3 Triangles and F are similar. alculate the length of a F b. 3. 5.2 cm F In uestions 4 7, is parallel to. and are straight lines. 4 5 2. 3. 2 cm 5. 10. alculate the length of a b. alculate the length of a b. 6 7 4. 1 9. 1. 6. alculate the length of a b. alculate the length of a b. 534
33.2 imilar polygons HTR 33 In uestions 8 10, is parallel to. and are straight lines. 8 9 2. 5.7 cm 3. 6. alculate the length of a b. alculate the length of a b. 10 3. alculate the length of a b. 33.2 imilar polygons Two polygons are similar if the lengths of pairs of corresponding sides are in the same proportion. For example, a square of side 2 cm and a square of side are similar because they have equal angles and corresponding sides are in the proportion 6 2 3 The same argument applies to any pair of squares and so all squares are similar. It also apples to any pair of regular polygons with the same number of sides. o, for example, all regular hexagons are similar. (ll circles are also similar.) Two rectangles may be similar. For example, rectangle R and rectangle are similar because 1 0 5 4 2 2 1 2 and 1 5 5 6 2 2 1 2. o, corresponding sides are in the same proportion. R 1 nother way of finding whether two rectangles are similar is to work out the value of l ength for width each of them. 10 cm For rectangle R, l eng wid th th 6 4 3 2 1 1 2 and for rectangle, l eng w id th th 1 5 1 0 3 2 1 1 2 The value of l ength is the same for both rectangles and so rectangles R and are similar. width This is not generally true for rectangles, however, as xample 4 shows. lthough any two rectangles have equal angles, this alone does not necessarily mean that they are similar. In this respect, shapes with more than three sides differ from triangles. 535
HTR 33 imilar shapes xample 4 how that rectangle and rectangle FGH are not similar. H G 9 cm 12 cm F olution 4 Method 1 F 1 2 1.5 8 F G 9 5 1.8 ivide the width of rectangle FGH by the width of rectangle. The lengths of pairs of corresponding sides are not in the same proportion and so the rectangles are not similar. Method 2 8 5 1.6 F 12 1.3 F G 9 ivide the length of rectangle FGH by the length of rectangle. Work out the value of l ength for rectangle. width Work out the value of l ength for rectangle FGH. width The value of l ength is different for each rectangle and so they are not similar. width The methods used to find the lengths of sides in pairs of similar triangles can be used to find the lengths of sides in other pairs of similar shapes. xample 5 entagons and are similar. alculate the value of a x b y. olution 5 a 8 6 x 4.8 x 8 4.8 6.4 6 4. The lengths of pairs of corresponding sides are in the same proportion. Rearrange the equation to work out the value of x. 7. b 8 6 7.6 y y 6 7.6 5.7 8 The lengths of pairs of corresponding sides are in the same proportion. Rearrange the equation to work out the value of y. 536
33.2 imilar polygons HTR 33 xercise 33 1 Rectangles and are similar. Work out the value of x. 2. 2 Rectangles R and are similar. Work out the value of y. R 5. 8. 10. 3 how that rectangles T and U are not similar. T U 10 cm 9 cm 1 4 re rectangles V and W similar? You must show working to explain your answer. V 4. W 6. 8. 5 photograph is 1 long and 10 cm wide. It is mounted on a rectangular piece of card so that there is a border wide all around the photograph. re the photograph and the card similar shapes? how working to explain your answer. 11.2 cm 1 10 cm 6 uadrilaterals and are similar. alculate the value of a x b y. 9 cm 7. 7 entagons R and are similar. alculate the value of a x b y. 3.2 cm 3. R 6.9 cm 8 Hexagons T and U are similar. alculate the value of a x b y. 2.7 cm 5.7 cm 4. 3.9 cm T U 537
HTR 33 imilar shapes 33.3 reas of similar shapes The diagram shows a square of side 1 cm and a square of side 2 cm. rea of the square of side 1 cm 1cm 2. rea of the square of side 2 cm 4cm 2. When lengths are multiplied by 2, area is multiplied by 4 The diagram shows a cube of side 1 cm and a cube of side 2 cm. urface area of the cube of side 1 cm 6 1cm 2 6cm 2. urface area of the cube of side 2 cm 6 4cm 2 2 2. gain, when lengths are multiplied by 2, area is multiplied by 4 The diagram shows a square of side 1 cm and a square of side. rea of the square of side 1 cm 1cm 2. rea of the square of side 9cm 2. When lengths are multiplied by 3, area is multiplied by 9 In general, for similar shapes, when lengths are multiplied by k,area is multiplied by k 2. For example, if the lengths of a shape are multiplied by 5, its area is multiplied by 5 2, that is 25 xample 6 uadrilaterals and are similar. The area of quadrilateral is 10 cm 2. alculate the area of quadrilateral. 12 cm olution 6 1 2 4 3 4 2 16 10 cm 2 16 160 cm 2 length of side in Work out to find the number by which lengths length of corresponding side in have been multiplied, that is, find the scale factor. quare the scale factor to find the number by which the area has to be multiplied. Multiply the area of quadrilateral by 16 to find the area of quadrilateral. xample 7 ylinders R and are similar. The surface area of cylinder R is 40 cm 2. alculate the surface area of cylinder. R 1 3 olution 7 3 5 14 2.5 2.5 2 6.25 Work out height of cylinder to find the number by which lengths have been height of cylinder R multiplied, that is, find the scale factor. quare the scale factor to find the number by which the area has to be multiplied. 40 cm 2 6.25 250 cm 2 Multiply the surface area of cylinder R by 6.25 to find the surface area of cylinder. 538
33.3 reas of similar shapes HTR 33 If the areas of two similar shapes are known, the scale factor can be found. For example, if two similar shapes T and U have areas 2 and 320 cm 2, the area of shape T has been multiplied by 32 0 64 5 o, if the scale factor is k, then k 2 64 and k 64 8 xample 8 entagons V and W are similar. The area of pentagon V is 40 cm 2 and the area of pentagon W is 90 cm 2. alculate the value of a x b y. olution 8 9 0 40 2.25 k 2 2.25 V W 1 Work out a rea of W to find the number by which the area has been multiplied. area of V This number is (scale factor) 2. k 2.25 1.5 a x 8 1.5 12 b y 1.5 18 18 y 12 1. 5 The scale factor is the square root of this number. Multiply the length on V by the scale factor to find the corresponding length on W. The length on V multiplied by the scale factor gives the corresponding length 1 on W. ivide 18 by the scale factor to find the value of y. xercise 33 1 uadrilaterals and are similar. The area of quadrilateral is 20 cm 2. alculate the area of quadrilateral. 1 2 Triangles and are similar. The area of triangle is 2. alculate the area of triangle. 2 cm 20 cm 3 entagons and are similar. The area of pentagon is 250 cm 2. alculate the area of pentagon. 20 cm 539
HTR 33 imilar shapes 4 ylinders and are similar. The surface area of cylinder is 60 cm 2. alculate the surface area of cylinder. 12 cm 1 5 uboids and are similar. The surface area of cuboid is 72 cm 2. alculate the surface area of cuboid. 6 ones and are similar. The surface area of cone is 6 2. alculate the surface area of cone. 20 cm 32 cm 7 arallelograms and are similar. The area of parallelogram is 36 times the area of parallelogram. alculate the value of x. 8 yramids and are similar. The surface area of pyramid is 64 times the surface area of pyramid. alculate the value of a x b y. 9 uadrilaterals and are similar. The area of quadrilateral is 10 cm 2. The area of quadrilateral is 360 cm 2. alculate the value of a x b y. 2 cm 40 cm 27 cm 10 ylinders and are similar. The surface area of cylinder is 50 cm 2. The surface area of cylinder is 72 cm 2. alculate the value of h. 1 h cm 11 Trapeziums and are similar. The area of trapezium is 3 2. The area of trapezium is 100 cm 2. alculate the value of a x b y. 8.1 cm 6. 540
33.4 Volumes of similar solids HTR 33 12 uboids and are similar. The surface area of cuboid is 50 cm 2. The surface area of cuboid is 162 cm 2. alculate the value of a x b y. 33.4 Volumes of similar solids The diagram shows a cube of side 1 cm and a cube of side 2 cm. Volume of the cube of side 1 cm (1 1 1) cm 3 1cm 3. Volume of the cube of side 2 cm (2 2 2) cm 3 8cm 3. When lengths are multiplied by 2, volume is multiplied by 8 The diagram shows a cube of side 1 cm and a cube of side. Volume of the cube of side 1 cm (1 1 1) cm 3 1cm 3. Volume of the cube of side 2 cm (3 3 3) cm 3 27 cm 3. When lengths are multiplied by 3, volume is multiplied by 27 In general, for similar solids, 3. when lengths are multiplied by k,volume is multiplied by k 3. 4. For example, if the lengths of a shape are multiplied by 5, its volume is multiplied by 5 3, that is 125. xample 9 uboids R and are similar. The volume of cuboid R is 50 cm 3. alculate the volume of cuboid. olution 9 2 4 4 6 R 2 length of side in Work out to find the number by which lengths length of corresponding side in R have been multiplied, that is, find the scale factor. 4 3 64 50 cm 3 64 3200 cm 3 ube the scale factor to find the number by which the volume has to be multiplied. Multiply the volume of cuboid R by 64 to find the volume of cuboid. xample 10 ylinders T and U are similar. The volume of cylinder T is 250 cm 3. The volume of cylinder U is 432 cm 3. alculate the value of h. olution 10 T 3 U h cm 4 32 1.728 Work out v olume of U to find the number by which the volume has been multiplied. 250 volume of T k 3 1.728 This number is (scale factor) 3. k 3 1.728 1.2 h 35 1.2 42 The scale factor is the cube root of this number. Multiply the height of cylinder T by the scale factor to find the height h cm of cylinder U. 541
HTR 33 imilar shapes xercise 33 1 uboids and are similar. The volume of cuboid is 20 cm 3. alculate the volume of cuboid. 2 ylinders and are similar. The volume of cylinder is 7 cm 3. alculate the volume of cylinder. 3 ones and are similar. The volume of cone is 40 cm 3. 12 cm 30 cm alculate the volume of cone. 4 risms and are similar. The volume of prism is 80 cm 3. alculate the volume of prism. 10 cm 5 yramids and are similar. The volume of pyramid is 320 cm 3. alculate the volume of pyramid. 12 cm 1 6 uboids and are similar. The volume of cuboid is 250 cm 3. 27 cm alculate the volume of cuboid. 1 7 pheres and are similar. The volume of sphere is 216 times the volume of sphere. alculate the value of d. d cm ll spheres are similar (so are all cuboids). 8 risms and are similar. The volume of prism is 1000 times the volume of prism. alculate the value of a x b y. 4 542
33.5 Lengths, areas and volumes of similar solids HTR 33 9 ylinders and are similar. The volume of prism is 30 cm 3. The volume of prism is 810 cm 3. alculate the value of h. h cm 10 yramids and are similar. The volume of pyramid is 40 cm 3. The volume of pyramid is 13 3. alculate the value of a x b y. 2 21 cm 11 uboids and are similar. The volume of cuboid is 12 3. The volume of cuboid is 729 cm 3. alculate the value of a x b y. 1 4 12 ones and are similar. The volume of cone is 128 cm 3. The volume of cone is 250πcm 3. alculate the value of x. 3 33.5 Lengths, areas and volumes of similar solids ometimes both area and volume are involved in questions on similar solids. xample 11 uboids R and are similar. The surface area of cuboid R is 60 cm 2. The surface area of cuboid is 1500 cm 2. The volume of cuboid R is 40 cm 3. alculate the volume of cuboid. R olution 11 15 00 25 60 k 2 25 k 25 5 5 3 125 40 cm 3 125 5000 cm 3 area of Work out to find the number by which the area has been multiplied. a rea of R This number is (scale factor) 2. The scale factor is the square root of this number. ube the scale factor to find the number by which the volume has to be multiplied. Multiply the volume of cuboid R by 125 to find the volume of cuboid. 543
HTR 33 imilar shapes xample 12 ylinders T and U are similar. The volume of cylinder U is 512 times the volume of cylinder T. The surface area of cylinder U is 1600 cm 2. alculate the surface area of cylinder T. olution 12 k 3 512 The number by which the volume has been multiplied is (scale factor) 3. T U k 3 512 8 8 2 64 1600 cm 2 64 2 2 The scale factor is the cube root of this number. quare the scale factor. o the surface area of cylinder U is 64 times the surface area of cylinder T. ivide the surface area of cylinder U by 64 to find the surface area of cylinder T. xercise 33 1 uboids and are similar. The surface area of cuboid is 25 times the surface area of cuboid. The volume of cuboid is 40 cm 3. alculate the volume of cuboid. 2 ylinders and are similar. The volume of cylinder is 27 times the volume of cylinder. The surface area of cylinder is 20 cm 2. alculate the surface area of cylinder. 3 ones and are similar. The surface area of cone is 2 2. The surface area of cone is 5 2. The volume of cone is 1 3. alculate the volume of cone. 4 risms and are similar. The volume of prism is 250 cm 3. The volume of prism is 68 3. The surface area of prism is 300 cm 2. alculate the surface area of prism. 5 yramids and are similar. The surface area of pyramid is 10 2. The surface area of pyramid is 300 cm 2. The volume of pyramid is 37 3. 7.2 cm alculate a the value of x b the volume of pyramid. 544
hapter summary HTR 33 6 uboids and are similar. The volume of cuboid is 2 3. The volume of cuboid is 37 3. The surface area of cuboid is 5 2. alculate a the value of x b the value of y c the surface area of cuboid. 3. 11. 7 ylinders and are similar. The surface area of cylinder is 90 cm 2. The surface area of cylinder is 250 cm 2. The volume of cylinder is 37 3. 3.9 cm d cm alculate a the value of d b the volume of cylinder. 8 yramids and are similar. The volume of pyramid is 1000 cm 3. The volume of pyramid is 172 3. The surface area of pyramid is 450 cm 2. alculate a the value of x b the surface area of pyramid. 17. 9 ones and are similar. The surface area of cone is 10 cm 2. The surface area of cone is 160 cm 2. The volume of cone is 6 cm 3. alculate the volume of cone. Give you answer as a multiple of. 10 Frustums and are similar. The volume of frustum is 256 cm 3. The volume of frustum is 500 cm 3. The surface area of frustum is 225 cm 2. alculate the surface area of frustum. Give your answer as a multiple of. hapter summary You should now know that: triangles which have the same shape but not the same size are called similar triangles similar triangles have equal angles all squares are similar and all circles are similar, as are all cubes and all spheres all regular polygons with the same number of sides are similar for similar shapes, the lengths of pairs of corresponding sides are in the same proportion for similar shapes, when lengths are multiplied by k,area is multiplied by k 2 for similar shapes, when lengths are multiplied by k,volume is multiplied by k 3. 545
HTR 33 imilar shapes hapter 33 review questions 1 Triangles and F are similar. 2. 2 cm 2. 2cm 1. F, angle F 49 a Find the size of angle. b Work out the length of i F ii. (4400 May 2005) 49 1. F iagrams NOT 2 In the triangle : is parallel to,, iagram NOT, 9 cm. a Work out the length of. 9 cm b Work out the length of. (1385 November 2001) 3 and are straight lines. is parallel to.,, 4.. Work out the length of. 4. iagram NOT 4 is parallel to. The lines and intersect at point O. 11 cm, O, O. 11 cm alculate the length of. O iagram NOT (1385 June 2001) 5 : 1:3 a Work out the length of. b Work out the length of. iagram NOT (1385 June 1998) 546
hapter 33 review questions HTR 33 6 ictures NOT 20 uro note is a rectangle 133 mm long and 72 mm wide. 500 uro note is a rectangle 160 mm long and 82 mm wide. how that the two rectangles are not mathematically similar. (1387 June 2004) 7 hapes and FGH are mathematically similar. 100 F 100 G iagrams NOT H a alculate the length of. b alculate the length of F. (1387 November 2003) 8 uadrilateral is mathematically similar to quadrilateral. 1 iagrams NOT 12 cm a alculate the value of x. b alculate the value of y. The area of quadrilateral is 60 cm 2. c alculate the area of quadrilateral. (4400 May 2004) 9 ylinder and cylinder are mathematically similar. The length of cylinder is and the length of cylinder is. The volume of cylinder is 80 cm 3. iagrams NOT alculate the volume of cylinder. (1387 November 2003) 547
HTR 33 imilar shapes 10 Two cuboids, and T,are mathematically similar. The total surface area of cuboid is 157 cm 2 and the total surface area of cuboid T is 2512 cm 2. T 2 iagrams NOT a The length of cuboid T is 2. alculate the length of cuboid. b The volume of cuboid is 130 cm 3. alculate the volume of cuboid T. (4400 May 2005) 11 solid plastic toy is made in the shape of a cylinder which is joined to a hemisphere at both ends. The diameter of the toy at the joins is. The length of the toy is 1. a alculate the volume of plastic needed to make the toy. Give your answer correct to three significant figures. 10 cm iagram NOT similar toy has a volume of 5500 cm 3. b alculate the diameter of this toy. Give your answer correct to three significant figures. (1385 November 1998) 12 is a right-angled triangle. is parallel to. 7., 2., 2.. a alculate the area of trapezium.. b alculate the value of x. The smallest angle of the trapezium is. The lengths shown in the diagram are correct to the nearest millimetre. c alculate the least possible value of tan. 2. 2. 7. iagram NOT θ (1385 June 1999) 13 solid statue is contained within a hemisphere of diameter 500 cm. a alculate the total surface area of the hemisphere, including the base. Give your answer in m 2, correct to three significant figures. The solid statue has a height of 80 cm and a mass of 1.5 kg. larger solid statue is geometrically similar and is made of the same material. It has a height of 12. 500 cm iagram NOT b alculate the mass of the larger solid statue. Give your answer in kg, correct to three significant figures. (1385 June 1999) 548
hapter 33 review questions HTR 33 14 The diagram represents a prism. It has a crosssection in the shape of a sector O of a circle, centre O.The radius of the sector is and the length of the prism is 12 cm. ngle O 54. a alculate i the area of sector O ii the total surface area of the prism. geometrically similar prism is made of the same material as the original prism and has a mass 64 times the mass of the original prism. b alculate the length of the heavier prism. (1384 November 1995) 54 O 12 cm iagram NOT 15 The diagram shows a frustum. The diameter of the base is 3d cm and the diameter of the top is d cm. The height of the frustum is h cm. The formula for the curved surface area, cm 2, of the frustum is 2 d h 2 d 2 a Rearrange the formula to make h the subject. Two mathematically similar frustums have heights of 20 cm and 30 cm. The surface area of the smaller frustum is 450 cm 2. b alculate the surface area of the larger frustum. (1387 June 2003) d cm 3d cm h cm iagram NOT 549