Similar Triangles Handout

Transcription

1 Similar riangles Handout When an architect draws plans for a house or an engineer makes a drawing of a machine part, the result is drawn to scale showing the same objects in reduced sizes. wo figures with the same shape are said to be similar. he silhouettes of the scales shown in Figure 1 are similar. FIU 1 efinition of Similar Figures wo figures are similar if one of the figures can be uniformly enlarged so that it is identical with the other figure. his just means that if two figures are similar then one can be blown up to match the other. he term similar may be applied to three-dimensional objects as well as plane objects. he two boes in Figure 2 are similar. FIU 2 In this handout, we will be concerned with similar triangles. You could probably guess that two triangles will have the same shape if three angles of one triangle have the same measure as three angles of the other triangle. heorem If two angles of one triangle have the same measure as two angles of another triangle, then the triangles are similar. c b f a e d F In Figure 3, Δ is similar to ΔF since = and = F. In symbols, we write Δ ~ ΔF. It is important to write the names of the triangles so that the letters of the corresponding angles are in the same order. he benefit that comes from having similar triangles is found in the following result. FIU 3 Note: he arcs drawn in the angles indicate which angles are equal. he same letter names the angle and the side opposite from it. heorem If two triangles are similar, then their corresponding sides are in proportion. For the triangles in Figure 3, this means that a d = b e = c f. his is called an etended proportion since it involves three fractions.

2 2 Similar riangles Handout XL 1 eferring to the triangles shown below, (a) name the similar triangles, (b) write an etended proportion that is true for these triangles and (c) if r =, n = 3, and t = 8, then find m. t s S r (a) Note from the figure that = N and =. hus S = and ΔS ~ ΔN. r (b) omparing the corresponding sides gives: n = s p = t m. Note that the corresponding sides r and n are opposite from the equal angles and N. (c) Using the proportion from part (b) we have r n = t m 3 = 8 m = 2 m = 6. m N p m n onsider the following eample. XL 2 he shadow of a flagpole has length 20 feet. t the same time, the length of the shadow of a yardstick is 6 inches. Find the height of the flagpole. F Y K If we assume that the flagpole and yardstick are perpendicular to the ground and that the rays from the sun are parallel, then F = Y and = K, and we have ΔF ~ ΔYK. his means that f y = g d = p k. We know that g = 20 ft, k = 1 yd = 3 ft and d = 6 in = 1 2 ft. Since p is the height of the flagpole, p 3 = 20 p = 0 p = he height of the flagpole is 120 feet.

3 Similar riangles Handout 3 In some situations, one similar triangle will overlap another. his usually means that one angle is shared by both triangles. XL 3 man, 6 feet tall, is walking away from a street light. If the length of the man s shadow is feet when he is 8 feet from the light, then how high is the light? N S 8 6 Let segment S represent the street light and N represent the man. Note that S = and that =. hus ΔS ~ ΔN. his means that s m = t n = S. Now we know that t = 12, n = N and N = 6. hus, 12 = S 6 S = 18. he street light is 18 feet high. here are certain problems in which the unknown side of one triangle may also be part of a side of the other triangle. XL Solve for Note that Δ ~ Δ and that the length of side = +. hus, + = 6 10 = = = 6.

4 m 1 m 2 Similar riangles Handout We know that when two triangles are similar, the corresponding sides are in proportion. In fact, there are many other corresponding parts of the triangles that are in the same proportion. wo of these are shown below. For eample, suppose that Δ ~ ΔU. U hen r b = a u = t g. n altitude of a triangle is the segment from a verte perpendicular to the side opposite that verte. Let h 1 be the altitude of Δ from and h 2 be the altitude of ΔU from U. U h h 1 h 2 Note: In some triangles, an altitude may lie outside of the triangle. hen h 1 h 2 = r b = a u = t g. median of a triangle is the segment joining a verte to the midpoint of the side opposite that verte. Let m 1 be the median of Δ from and m 2 be the median of ΔU from. U hen m 1 m 2 = r b = a u = t g he study of similar triangles has many practical applications. etermining distances that are difficult or impossible to measure is one such application. his is how the distance between the arth and oon was first found. In addition, similar triangles are the basis for right triangle trigonometry and its many important applications.

5 Similar riangles Handout 5 XISS NO: HS OLS O ON ON NININ. For eercises 1, state whether the two triangles are always, sometimes, or never similar. 7. Name the similar triangles in the figure. I 1. wo equilateral triangles. 2. wo right triangles. O 3. right triangle and an equilateral triangle.. wo right triangles, one with an acute angle of measure 60 and the other with an acute angle of measure Name the similar triangles in the figure. [Hint: here are three.] 5. onsider the triangles shown below. a) Name the similar triangles. N 9. flagpole casts a shadow of 28 feet at the same time the shadow of a person 6 feet tall is 2 feet long. How tall is the flagpole? b) Write an etended proportion that is true for these triangles. c) If t = 2, a = 3, and n = 6, find e. 6. onsider the triangles shown below. 10. he reek mathematician hales was known to have calculated the height of the great pyramid of heops by measuring the shadow of a pole. He found that the base of the pyramid has sides that are 756 ft. long. If his 3 ft. pole cast a shadow that was 6 ft. long at the same time that the shadow of the pyramid was 586 ft. long, determine the height of the pyramid to the nearest foot. a) Name the similar triangles. h b) Write an etended proportion that is true for these triangles. 756 ft. 586 ft. shadow c) If g = 30, = 36, and t = 5, find.

6 6 Similar riangles Handout 11. o measure the distance between points and on two opposite sides of a canyon, a man takes the following measurements: = 5 m, = 6 m, and = 15 m. What is the distance between the sides of the canyon? 12. girl, 5 feet tall, notices that when she is 2 feet from her boyfriend the line of sight to the top of a building just passes over the top of his head. Her boyfriend is 6 feet tall and is standing 100 feet from the building. How tall is the building? 13. Find in the figure below Find H in the figure below. H 6 O 15. Find the value of each variable in the figure below. 2 y Find the value of each variable in the figure below. + y y y 17. Wires are stretched from the top of each of two vertical poles to the bottom of the other as shown. If one pole is feet tall and the other 12 feet tall, how far above the ground do the wires cross? h a b [Hint: Look for two pairs of similar triangles.] For ercises 18 and 19, use the triangles shown below. J 60 h h a) Find the missing side lengths. b) Find the ratio of the perimeter of ΔJ to the perimeter of Δ. 19. Suppose that h 1 = 8. a) Find h 2. b) Find the ratio of the area of ΔJ to the area of Δ. 20. Suppose that Δ ~ ΔF and let h 1 be the altitude from and h 2 be the altitude from. Find the ratio of the area of Δ to the area of ΔF in terms of a and d.

7 7 Similar riangles Handout nswers to odd numbered eercises. 1. lways 3. Never 5. a) Δ ~ ΔN b) b m = a e = t n c) 9 7. ΔI ~ ΔO 9. 8 feet 11. = 75 m 13. = = 20, y = h = a) h 2 = 32 b) 9