31. Archaeology To estimate the height of a stone figure, Anya holds a small square up to her eyes and walks backward from the figure. She stops when the bottom of the figure aligns with the bottom edge of the square and the top of the figure aligns with the top edge of the square. Her eye level is 1.84 m from the ground. She is 3.50 m from the figure. What is the height of the figure to the nearest hundredth of a meter?

    

32. Reasoning Suppose the altitude to the hypotenuse of a right triangle bisects the hypotenuse. How does the length of the altitude compare with the lengths of the segments of the hypotenuse? Explain.

The diagram shows the parts of a right triangle with an altitude to the hypotenuse. For the two given measures, find the other four. Use simplest radical form.

Image Long Description

33. h = 2 , h equals 2 comma   s 1 = 1 s sub 1 , equals 1
34. a = 6 , eh equals 6 comma   s 1 = 6 s sub 1 , equals 6
35. l 1 = 2 , l sub 1 . equals 2 comma   s 2 = 3 s sub 2 , equals 3
36. s 1 = 3 , s sub 1 . equals 3 comma   l 2 = 6 3 l sub 2 , equals , 6 square root of 3
37. Coordinate Geometry C D ¯ c d bar  is the altitude to the hypotenuse of right △ A B C . white up pointing triangle eh b c .  The coordinates of A, D, and B are (4, 2), (4, 6), and (4, 15), respectively. Find all possible coordinates of point C.

Algebra Find the value of x.

Use the figure below for Exercises 42 and 43.

42. proof Prove Corollary 1 to Theorem 7-3.
    Given: Right △ A B C white up pointing triangle eh b c  with altitude to the hypotenuse C D ¯ c d bar
    Prove: A D C D = C D D B fraction eh d , over c d end fraction . equals . fraction c d , over d b end fraction
43. proof Prove Corollary 2 to Theorem 7-3.
    Given: Right △ A B C white up pointing triangle eh b c  with altitude to the hypotenuse C D ¯ c d bar
    Prove: A B A C = A C A D ′ A B B C = B C D B fraction eh b , over eh c end fraction . equals . fraction eh c , over eh , d prime end fraction . fraction eh b , over b c end fraction . equals . fraction b c , over d b end fraction

C challenge

44. proof Given: Right △ A B D white up pointing triangle eh b d  with altitude to the hypotenuse B E ¯ , b e bar , comma  and equilateral △ A B C white up pointing triangle eh b c
    Prove: B E = A E 3 b e equals eh e square root of 3
    
45. 1. Consider the following conjecture: The product of the lengths of the two legs of a right triangle is equal to the product of the lengths of the hypotenuse and the altitude to the hypotenuse. Draw a figure for the conjecture. Write the Given information and what you are to Prove.
    2. Reasoning Is the conjecture true? Explain.

Table of Contents