Mathematics SK, Strand H UNIT H4 ongruence and Similarity: Text STRN H: ngle Geometry H4 ongruency and Similarity Text ontents Section * * H4.1 ongruence H4. Similarity IMT, Plymouth University
Mathematics SK, Strand H4 ongruence and Similarity: Text H4 ongruence and Similarity H4.1 ongruence Two shapes are said to be congruent if they are the same shape and size: that is, the corresponding sides of both shapes are the same length and corresponding angles are the same. cm The two triangles shown here are congruent. 7 cm 7 cm cm Shapes which are of different sizes but which have the same shape are said to be similar. The triangle below is similar to the triangles above but because it is a different size it is not congruent to the triangles above... 1. There are four tests for congruence which are outlined below. TST 1 (Side, Side, Side) If all three sides of one triangle are the same as the lengths of the sides of the second triangle, then the two triangles are congruent. This test is referred to as SSS. TST (Side, ngle, Side) If two sides of one triangle are the same length as two sides of the other triangle and the angle between these two sides is the same in both triangles, then the triangles are congruent. This test is referred to as SS. IMT, Plymouth University 1
H4.1 Mathematics SK, Strand H4 ongruence and Similarity: Text TST (ngle, ngle, Side) If two angles and the length of one corresponding side are the same in both triangles, then they are congruent. This test is referred to as S. TST 4 (Right angle, Hypotenuse, Side) If both triangles contain a right angle, have hypotenuses of the same length and one other side of the same length, then they are congruent. This test is referred to as RHS. Worked xample 1 Which of the triangles below are congruent to the triangle, and why? F 45.5 4 cm.. 5.4 4 cm H L 5.4 4 cm I 45.5 G J. 5.4 K Solution onsider first the triangle F: = F = = F s the sides lengths are the same in both triangles the triangles are congruent. (SSS) IMT, Plymouth University
H4.1 Mathematics SK, Strand H4 ongruence and Similarity: Text onsider the triangle GHI: = HI ˆ = GHI ˆ ˆ = G IH ˆ s the triangles have one side and two angles the same, they are congruent. (S) onsider the triangle JKL: Two sides are known but the angle between them is unknown, so there is insufficient information to show that the triangles are congruent. Worked xample F is a square and = F. Find the pairs of congruent triangles in the diagram. Solution G onsider the triangles and F: F ˆ = F (F is a square.) = F (This is given in the question.) = F ˆ = 90 (They are corners of a square.) The triangles and F have two sides of the same length and also have the same angle between them, so these triangles are congruent. (SS) onsider the triangles G and G: = ( and F are congruent.) G = G (They are the same line.) G ˆ = G ˆ = 90 (This is given in the question.) oth triangles contain right angles, have the same length hypotenuse and one other side of the same length. So the triangles are congruent. (RHS) Investigation 1. How many straight lines can you draw to divide a square into two congruent parts?. How many lines can you draw to divide a rectangle into two congruent parts?. an you draw two straight lines through a square to divide it into four congruent quadrilaterals which are not parallelograms? IMT, Plymouth University
H4.1 Mathematics SK, Strand H4 ongruence and Similarity: Text xercises 1. Identify the triangles below which are congruent and give the reasons why. F 8 cm 4 cm 8 cm 4 cm G 69 K 66 4.61 cm 4.61 cm 6.08 cm I 45 6.08 cm H J L O M 85 10 cm R 75 10 cm T 0 0 Q N P 4.61 cm S 45 U IMT, Plymouth University 4
H4.1 Mathematics SK, Strand H4 ongruence and Similarity: Text. If O is the centre of the circle, prove that the triangles O and O are congruent. O. O If O is the centre of both circles, prove that the triangles O and O are congruent. 4. If O is the centre of the circle, prove that the triangles O and O are congruent. O 5. F When = F, this rhombus contains two congruent triangles. Identify the triangles and prove that they are congruent. IMT, Plymouth University 5
H4.1 Mathematics SK, Strand H4 ongruence and Similarity: Text 6. If O is the centre of the circle and =, show that and are congruent triangles. O 7. Two triangles have sides of lengths 8 cm and, and contain an angle of 0. Show that it is possible to draw 4 different triangles, none of which are congruent, using this information. 8. The diagram shows the parallelogram with diagonals which intersect at X. Prove that the parallelogram contains pairs of congruent triangles. X H G 9. The diagram shows a cuboid. The mid-point of the side is X. Show that triangles HG and F are congruent. X Show that triangles X and G are congruent. F 4 Investigation 1. In how many ways can you cut a square-based cake into two congruent parts?. How can you use eight straight lines of equal length to make a square and four congruent equilateral triangles? hallenge! Study the diagram opposite and then find the ratio of the area of the large triangle to that of the small triangle. 4 IMT, Plymouth University 6
Mathematics SK, Strand H4 ongruence and Similarity: Text H4. Similarity Similar shapes have the same shape but may be different sizes. The two rectangles shown below are similar they have the same shape but one is smaller than the other. cm cm They are similar because they are both rectangles and the sides of the larger rectangle are three times longer than the sides of the smaller rectangle. It is interesting to compare the area of the two rectangles. The area of the smaller rectangle is and the area of the larger rectangle is 54 cm, which is nine times ( ) greater. Note In general, if the lengths of the sides of a shape are increased by a factor k, then the area is increased by a factor k. 9 cm These two triangles are not similar. The sides lengths of the triangles are not in the same ratio and so the triangles are not similar. 4 5 4 6 5 For two triangles to be similar, they must have the same internal angles, as shown in the similar shapes below. 100 7.84 50 7.71 0 1 50.9 100 6 15.4 0 IMT, Plymouth University 7
H4. Mathematics SK, Strand H4 ongruence and Similarity: Text The diagrams below show similar cubes. 1 cm cm cm 1 cm 1 cm cm cm cm cm Length rea of Surface of side one face area Volume 1 cm 1 cm 1 cm cm 4 cm 4 cm 8 cm cm 9 cm 54 cm 7 cm The table gives the lengths of sides, area of one face, total surface area and volume. omparing the larger cube with the 1 cm cube we can note that: For the cm cube For the cm cube The lengths are times greater. The areas are 4 = times greater. The volume is 8 = times greater. The lengths are times greater. The areas are 9 = times greater. The volume is 7 = times greater. Note If the lengths of a solid are increased by a factor, k, its surface area will increase by a factor k and its volume will increase by a factor k. Worked xample 1 Which of the triangles,,,,, shown below are similar? 6.1 cm 70 8 cm 1. 70 1 60 50 9.04 cm 60 18.08 cm 50 IMT, Plymouth University 8
H4. Mathematics SK, Strand H4 ongruence and Similarity: Text 50. 4 cm 60 4. 70 How do the areas of the triangles which are similar compare? Solution First compare triangles and. Here all the lengths of the sides are twice the length of the sides of triangle, so the two triangles are similar. Then compare triangles and. Here all the angles are the same in both triangles, so the triangles must be similar. Finally, compare triangles and. 1 1 1 Note that 4 = 8 and 45. = 904., but 5. 61.. So these triangles are not similar. The lengths of the sides of triangle are times greater than the lengths of the sides of triangle, so the area will be = 4 times greater. The side lengths of triangle are of the side lengths of triangle. 4 So the area will be 4 9 = of the area of triangle. 16 The ratio of the areas of triangles : : can be written as: 9 16 : 1: 4 or 9: 16: 64 Worked xample xplain why triangles and are similar. Find the lengths of x and y. x 6 (c) Solution Find the ratio of the area of to. 4 6 y s the lines and are parallel, ˆ = ˆ 9 and ˆ = ˆ IMT, Plymouth University 9
H4. Mathematics SK, Strand H4 ongruence and Similarity: Text s the vertex, is common to both triangles, ˆ = ˆ So the three angles are the same in both triangles and therefore they are similar. omparing the sides and, the lengths in the larger triangle are 1.5 times the lengths in the smaller triangle. lternatively, it can be stated that the ratio of the lengths is :. So the length will be 1.5 times the length. = 15. 6 = 9 So y = In the same way, = 15., so 4 + x = 15. x 4 = 05. x x = 8 (c) s the lengths are increased by a factor of 1.5 or for the larger triangle, the areas will be increased by a factor of 1. 5 or. We can say that the ratio of the areas of the triangles is 1 :.5 or 1 : 9 4 or 4 : 9. If the area of triangle is 4k, then the area of triangle is 9k and hence the area of the quadrilateral is 5k. So the ratio of the area of to is 4 : 5. Worked xample The diagrams show two similar triangles. cm 4 cm F If the area of triangle F is 6.4, find the lengths of its sides. IMT, Plymouth University 10
H4. Mathematics SK, Strand H4 ongruence and Similarity: Text Solution If the lengths of the sides of triangle F are a factor k greater than the lengths of the sides of triangle, then its area will be a factor k greater than the area of. 1 rea of = 4 = 6 cm. So 6 k = 646. k = 441. k = 441. = 1. So the lengths of the sides of triangle F will be.1 times greater than the lengths of the sides of triangle. Worked xample 4 = 1. 5 = 10. F = 1. = 6. cm F = 1. 4 = 8. cm can has a height of 10 cm and has a volume of 00 cm. can with a similar shape has a height of 1 cm. Find the volume of the larger can. Find the height of a similar can with a volume of 675 Solution cm. The lengths are increased by a factor of 1., so the volume will be increased by a factor of 1.. Volume = 00 1. = 45. If the lengths are increased by a factor of k, then the volume will be increased by k. 675 = 00 k k =. 75 k =. 75 = 15. IMT, Plymouth University 11
H4. Mathematics SK, Strand H4 ongruence and Similarity: Text So the height must be increased by a factor of 1.5, to give xercises height = 15. 10 = 1 1. Which of the triangles below are similar? iagrams are not drawn to scale. 57.1 7 cm 78.5 44.4 4 cm 60 8 cm 6.9 cm 0 1 cm 60 0 cm 11 cm 0 14 cm F 44.4 5 1 cm 48 cm 78.5 57.1 44 cm. The diagram shows similar triangles. 1.99 4. 60 5 F opy the triangles and label all the angles and the lengths of all the sides. How do the areas of the two triangles compare? (xpress your answer as a ratio.) IMT, Plymouth University 1
H4. Mathematics SK, Strand H4 ongruence and Similarity: Text. The diagram shows two similar rectangles, and FGH. F cm H G Find the lengths of and H if the ratio of the area of to the area of FGH is: 1 : 4 4 : 9. 4. The diagram shows two regular hexagons and =. What is the ratio of the area of the smaller hexagon to the area of the larger hexagon? What is the ratio of the area of the smaller hexagon to that of the shaded area? 5. xplain why the triangles and are similar. G H I K J F Find the lengths of: 7 cm (i) (ii) 11 (iii) cm (c) Find the ratio of the areas of: (i) to (ii) to. 11 6. xplain why and are similar triangles. 49.5 What can be deduced about the lines and? 7. cm (c) Find the lengths x and y. (d) Find the ratio of the area of the triangle to the area of the quadrilateral.. 4 cm x 108. y IMT, Plymouth University 1
H4. Mathematics SK, Strand H4 ongruence and Similarity: Text 7. bottle has a height of 8 cm and a volume of 0 cm. Find the volume of similar bottles of heights: 1 cm 10 cm (c) 0 cm. 8. box has a volume of 50 cm and a width of. similar box has a width of 1 cm. Find the volume of the larger box. How many times bigger is the surface area of the larger box? 9. packet has the dimensions shown in the diagram. ll the dimensions are increased by 0%. Find the percentage increase in: (i) surface area (ii) volume. Find the percentage increase needed in the dimensions of the packet to increase the volume by 50%. 0 cm RISPY ITS for TS 0 cm 8 cm 10. Two similar cans have volumes of 400 cm and 150 cm. Find the ratio of the heights of the cans. Find the ratio of the surface areas of the cans. 11. One box has a surface area of 96 has a volume of 178 cm and a height of 4 cm. second similar box cm and a surface area of 864 cm. Find: the height of the larger box the volume of the smaller box. 1. X 18.9 cm iagram not accurately drawn 4. O 1 Y alculate the length of OY. alculate the size of angle XOY. IMT, Plymouth University 14
H4. Mathematics SK, Strand H4 ongruence and Similarity: Text 1. Triangles, PQR and HIJ are all similar. J Not to scale R 0 10 cm 1. Not to P Q H I alculate the length of. What is the size of angle? (c) alculate the length of HJ. 14. m I stood 40 m away from the tallest building in Singapore. I held a piece of wood 40 cm long at arms length, 60 cm away from my eye. The piece of wood, held vertically, just blocked the building from my view. Use similar triangles to calculate the height, h metres, of the building. 15. = 4 cm, =, angle = 5. is perpendicular to. Not to scale alculate. 5 alculate angle. 4 cm (c) Triangle is similar to triangle. The area of triangle is nine times the area of triangle. (i) What is the size of angle? (ii) Work out the length of. N IMT, Plymouth University 15
H4. Mathematics SK, Strand H4 ongruence and Similarity: Text 16. roof has a symmetrical frame, with dimensions as shown. = PR = P T ˆ = 90 5 m Not to scale R S 5 m P T 18 m Q (i) Write down a triangle which is similar to triangle. (ii) alculate the length PR. alculate the value of angle T. 17. Two wine bottles have similar shapes. The standard bottle has a height of 0 cm. The small bottle has a height of.. Not to scale 0 cm. Standard bottle Small bottle (c) alculate the ratio of the areas of the bases of the two bottles. Give your answer in the form n : 1. What is the ratio of the volumes of the two bottles? Give your answer in the form n : 1. Is it a fair description to call the small bottle a 'half bottle'? Give a reason for your answer. IMT, Plymouth University 16
H4. Mathematics SK, Strand H4 ongruence and Similarity: Text 18. The normal size and selling price of small and medium toothpaste is shown. SMLL SIZ 60 ml 50 cents MIUM SIZ 15 ml $1 supermarket sells the toothpaste on special offer. (c) The special offer small size has 0% more toothpaste for the same price. How much more toothpaste does it contain? The special offer medium size costs 90 cents for 15 ml. What is the special offer price as a fraction of the normal price? alculate the number of ml per cent for each of these special offers. Which of these special offers gives better value for money? You must show your working. (d) (i) The 60 ml content of the small size has been given to the nearest 10 ml. What is the smallest number of ml it can contain? (ii) The 15 ml content of the medium size has been given to the nearest 5 ml. What is the smallest number of ml it can contain? 19. Two bottles of perfume are similar to each other. The heights of the bottles are 4 cm and. The smaller bottle has a volume of 4 cm. alculate the volume of the larger bottle. Two bottles of aftershave are similar to each other. The areas of the bases of these bottles are 4.8 cm and 10.8 cm. The height of the smaller bottle is cm. alculate the height of the larger bottle. IMT, Plymouth University 17