4.3 Similar riangles Focus on fter this lesson, you will be able to determine similar triangles determine if diagrams are proportional solve problems using the properties of similar triangles S 3 Students and Staff Materials tracing paper ruler protractor corresponding angles corresponding sides have the same relative position in geometric figures corresponding angles: and and and F corresponding sides: and and F and F F onnie and Justin created these logos for the Student ouncil. heir advisor tells them that the triangles are similar. How can she tell? hat do you know about similar figures? hat strategies can you develop to determine if triangles are similar? plore How to Identify Similar riangles 1. race each logo on separate pieces of tracing paper.. a) Measure the angles in each logo. hat do you notice about the corresponding angles? b) Measure the side lengths in each logo. hat do you notice about the ratios of the corresponding sides of the triangles? Reflect and heck 3. a) hat conclusions can you make about the corresponding angles of the two triangles? b) hat conclusions can you make about the corresponding sides of the two triangles? 4. a) hat conditions do you think are necessary in order for two triangles to be similar? b) est the conditions on a different set of two triangles. re the triangles similar? iscuss with a classmate why you think the triangles are, or are not, similar. 146 MHR hapter 4
ink the Ideas ample 1: Identify Similar riangles etermine if is similar to FG. 1 37 4 F 37 5 3 9 15 G similar (figures) have the same shape but different size have equal corresponding angles and proportional corresponding sides Solution Similar triangles have corresponding angles that are equal in measure and corresponding sides that are proportional in length. ompare corresponding angles: = 90 and = 90 = 37 and F = 37 = 53 and G = 53 he corresponding angles are equal. ompare corresponding sides: F = 1 4 FG = 15 5 G = 9 3 = 3 = 3 = 3 he sum of the angles in a triangle is 180. If you know the measures of two pairs of angles are equal, then what can you conclude about the third pair of angles? he corresponding sides are proportional with a scale factor of 3. FG iteracy ink he symbol means is similar to. FG means triangle is similar to triangle FG. iteracy ink ngles can be named in two ways: Use three capital letters. he middle letter is the verte of the angle. Use only the middle letter identifying the verte. Use a single letter when there is only one angle at a verte. For eample, the angle at verte can be named M or. M 4.3 Similar riangles MHR 147
eb ink o learn more about properties of similar triangles, go to www. mathlinks9.ca and follow the links. etermine if each pair of triangles is similar. Show how you know. a) b) Show You now Q 1.1 69 R.7 3.3.8 P 4. 3 70 3 4.5 4.5 70 F 4.1 S 1.7 69 Strategies Organize, nalyse, and Solve ample : Use Similar riangles to etermine a Missing Side ength yle is drawing triangles for a math puzzle. Use your knowledge of similar triangles to determine a) if the triangles are similar b) the missing side length Solution a) heck that M is similar to UV. he sum of the angles in a triangle is 180. = 180-50 - 85 = 45 U = 180-85 - 45 = 50 ompare corresponding angles: = 45 and = 45 = 50 and U = 50 M = 85 and V = 85 ll pairs of corresponding angles are equal. herefore, M UV. 50 85 4 1 M 45 7 85 U V 8 It is not necessary to prove both conditions for similarity. One is sufficient. b) You can compare corresponding sides to determine the scale factor. M UV = 4 M 8 V = 1 7 U = = 3 = 3 = he scale factor is 3. You can solve for the unknown length. 148 MHR hapter 4
Method 1: Use a Scale Factor Since the triangles are similar, you can use the scale factor to determine the missing length. = 3 = 31.5 he missing side length is 31.5 units. Method : Use a Proportion Since the triangles are similar, you can use equal ratios to set up a proportion. M V = U 1.5 1 7 = 1.5 = 31.5 he missing side length is 31.5 units. How would you solve for? 7 = 1.5 Show You now Solve using a method of your choice. a) GHI is similar to M. hat is the missing side length? press your answer to the nearest tenth. H 1.9 G I 10 8 7.6 M b) is similar to F. etermine the missing side length. press your answer to the nearest tenth. 8.7 5.4 7. F 8.7 4.3 Similar riangles MHR 149
ey Ideas riangles are similar if one of the following conditions holds true: corresponding angles are equal in measure corresponding sides are proportional in length F is similar to. F is not similar to PQR. =, =, F = 45 = 3 1.5 F =. 1.1 F =.6 1.3 = = = You can solve problems related to similar triangles using different methods. Use a scale factor. Use a proportion. 60 3.6. 75 F 1.5 45 1.3 60 75 1.1 P Q.8 R heck Your Understanding ommunicate the Ideas 1. If two triangles are similar, what can you say about the angles of the triangles? the side lengths of the triangles?. manda is unclear about similar triangles. She drew these two triangles and states they are similar. Is she correct? plain. X 5 3.1 3. re two triangles that have equal angles and equal sides similar? Use an eample to support your answer. Y 4.6 Z M.4 N Practise For help with #4 to #8, refer to ample 1 on page 147. 4. ist the corresponding angles and the corresponding sides for PQR and UV. 5. hat are the corresponding angles and the corresponding sides in this pair of triangles? Y P X Q R U V 150 MHR hapter 4
6. re the triangles similar? Show how you know. V S R.5 1.. 1.5 11 7. etermine if the triangles are similar. Show how you know. 4 6 U 6 For help with #9 to #11, refer to ample on pages 148 149. 9. SR is similar to UV. etermine the missing side length. S 8 R V 73.5 73.5 10. is similar to GF. hat is the missing side length? U 8 13.5.5 8.5 10 18 F 1.5 7.5 X 1 8. etermine which pairs of triangles are similar. Use a sketch to help eplain how you know. riangle ngles Sides = 90 = 45 = 45 FG = 90 F = 45 G = 45 HIJ H = 90 I = 60 J = 30 M = 90 = 45 M = 45 Y = 6 = 8.4 = 6 F = 3 FG = 4. G = 3 HI = 9. IJ = 18.4 HJ = 15.9 = 9 M = 1.6 M = 9 11. raw a triangle that is similar to the one shown. abel the measurements for angles and sides on your similar triangle. pply J 1. Sam built a ramp to a loading dock. he ramp has a vertical support m from the base of the loading dock and 3 m from the base of the ramp. If the vertical support is 1. m in height, what is the height of the loading dock? G 1. m 3 m m 4.3 Similar riangles MHR 151
13. wo etension ladders are leaning at the same angle against a vertical wall. he 3-m ladder reaches.4 m up the wall. How much farther up the.4 m wall does the 8-m ladder reach? 3 m 8 m tend 18. tourist wants to estimate the height of an office tower. He places a mirror on the ground and moves away to sight the top of the tower in the mirror. 14. rin, who is 1.60 m tall, casts a shadow that is 1.5 m long. Her shadow etends to the end of a tree s shadow when she stands 4.75 m from the tree. hat is the height of the tree? 19 cm 1.60 m 4.75 m 1.5 m 15. Sara was helping her father assemble a slide for the local park. He decides to reinforce the slide with an etra support beam. How long should the etra support beam be? support beam 195 cm 57 cm 15 cm 1 cm 16. Peter, who is 168 cm tall, casts a 45-cm shadow. Michael, who is standing beside him, casts a 40-cm shadow. an you tell who is taller? Use a diagram to help eplain why or why not. 17. evelop a word problem that can be solved using similar triangles. Include a diagram. 0.4 m 87.6 m a) How tall is the tower? b) In this situation, why is the mirror reflection a better way to indirectly measure the tower than by using shadows? 19. Is it possible for the two triangles described below to be similar? plain your reasoning. a) wo angles of one triangle measure 60 and 70. wo angles of the other triangle measure 50 and 80. b) wo angles of one triangle measure 45 and 75. wo angles of the other triangle measure 45 and 60. 0. he sides of a triangle measure 3 cm, 5 cm, and 6 cm. If the side of a similar triangle corresponding to 3 cm measures 8 cm, a) determine the lengths of the other sides b) determine the ratio of the perimeter of the smaller triangle to the perimeter of the larger triangle 1. Using a measuring tape, your shadow, and yourself, how can you determine the height of your school without actually measuring it? 15 MHR hapter 4
. XY is similar to ZY. alculate ZY to the nearest tenth. 3. Use two different sets of measurements to determine the area of M. X Z 10 cm 7 cm Y h 5 cm 0 cm M For your design project report, include a signature logo that features your name. a) On a sheet of 8.5 11 paper, design your logo. Include a triangle that is similar to the one shown. Measure all the angles and side lengths. b) raw a scale diagram of the logo to fit on your design project. Identify the scale factor you used. ech ink Similarity and Scale Factors In this activity, you can use dynamic geometry software to eplore similarity and scale factors. o use this activity, go to www.mathlinks9.ca and follow the links. plore 1. Slide point X along line segment and describe what happens to the image drawing.. How do the measures of the corresponding sides of the drawing change relative to each other? plain. 3. ompare the scale factor to the lengths of the sides of the original drawing and the image drawing. reate and complete a table similar to the one below with measurements taken at different locations. iscuss your findings with a classmate. Hint: In the table, m means the measure of. mf mf mx mx 4.3 Similar riangles MHR 153