36. Space Exploration The International Space
Station orbits at an altitude of about 250 miles
above Earth’s surface. The radius of Earth is
approximately 3963 miles. How far can an
astronaut in the space station see to the
horizon? Round to the nearest mile.
x mi
3963 mi
250 mi
37. Critical Thinking In the proof of the
Pythagorean Theorem on the first page of this
lesson, how do you know the outer figure is a
square? How do you know the inner figure is
a square?
Not drawn to scale
Multi-Step Find the perimeter and the area of each figure. Give your answer in
simplest radical form.
39.
38.
40.
8
8
6
17
12
8
8
12
41.
42.
5
43.
5
15
3
6
15
4
5
12
8
44. Write About It When you apply both the Pythagorean Theorem and its converse,
you use the equation a 2 + b 2 = c 2. Explain in your own words how the two theorems
are different.
B
Q
45. Use this plan to write a paragraph proof of the
Converse of the Pythagorean Theorem.
Given: ABC with a 2 + b 2 = c 2
Prove: ABC is a right triangle.
c
A
a
C
b
x
P
a
R
b
Plan: Draw PQR with ∠R as the right angle, leg lengths of a and b, and a hypotenuse
of length x. By the Pythagorean Theorem, a 2 + b 2 = x 2. Use substitution to
compare x and c. Show that ABC PQR and thus ∠C is a right angle.
46. Complete these steps to prove the Distance Formula.
K(x 2, y 2)
y
Given: J(x 1, y 1) and K(x 2, y 2) with x 1 ≠ x 2 and y 1 ≠ y 2
Prove: JK = √
(x 2 - x 1) 2 + (y 2 - y 1) 2
−−
a. Locate L so that JK is the hypotenuse of right JKL.
What are the coordinates of L?
b. Find JL and LK.
c. By the Pythagorean Theorem, JK 2 = JL 2 + LK 2. Find JK.
366
Chapter 5 Properties and Attributes of Triangles
x
L
S
500 mi
1300 mi
K
R
390 mi
M
Transtock Inc./Alamy Images
47. The figure shows an airline’s routes between four cities.
a. A traveler wants to go from Sanak (S) to Manitou (M).
To minimize the total number of miles traveled,
should she first fly to King City (K) or to Rice Lake (R)?
b. The airline decides to offer a direct flight from Sanak (S)
b
to Manitou (M). Given that the length of this flight is
more than 1360 mi, what can you say about m∠SRM?
J(x 1, y 1)