In , the altitudes of the triangle intersect at a point

We wish to determine x in Figure 5.56b so that the total distance is as small as possible. Consider three possible choices of this point on the river. These are suggested by dashed, dotted, and solid lines in Figure 5.57a. Also consider the reflections of the triangles across the river. [See Figure 5.57b.] D = d 1 + d 2 configuration with the solid line segments minimizes the distance. In that case, the triangle at left and the reflected triangle at right are similar. See Figure 5.58. 쮿 a 100 ft 200 ft Figure 5.56 d 1 x 200 b 100 450 d 2 a Figure 5.57 b d 1 x 200 100 d 2 100 450 – x 450 Figure 5.58 Thus Accordingly, the desired point on the river is 300 ft determined by upstream from the hiker’s location. 450 - x x = 150 300x = 45,000 200x = 45,000 - 100x 200x = 100450 - x 200 100 = 450 - x x The minimum distance D occurs where the segments of lengths d 1 and d 2 form a straight line. That is, the Summary A LOOK BACK AT CHAPTER 5 One goal of this chapter has been to define similarity for two polygons. We postulated a method for proving trian- gles similar and showed that proportions are a conse- quence of similar triangles, a line parallel to one side of a triangle, and a ray bisecting one angle of a triangle. The Pythagorean Theorem and its converse were proved. We discussed the 30°-60°-90° triangle, the 45°-45°-90° trian- gle, and other special right triangles with sides forming Pythagorean triples. The final section developed the con- cept segments divided proportionally. A LOOK AHEAD TO CHAPTER 6 In the next chapter, we will begin our work with the circle. Segments and lines of the circle will be defined, as will special angles in a circle. Several theorems dealing with the measurements of these angles and line segments will be proved. Our work with constructions will enable us to deal with the locus of points and the concurrence of lines that are found in Chapter 7. KEY CONCEPTS

5.1

Ratio • Rate • Proportion • Extremes • Means • Means- Extremes Property • Geometric Mean • Extended Ratio • Extended Proportion 5.2 Similar Polygons • Congruent Polygons • Corresponding Vertices, Angles, and Sides