Beginning Algebra, 11th Edition

One of the many proofs of the Pythagorean theorem was given in the Connections boxin this
section. Two more are given in Exercises 83 and 84.

83.Another proof of the Pythagorean theorem, attributed to the Hindu mathematician
Bhaskara, is based on the figures shown here. The figure on the left is made up of the
same square and triangles as the figure on the right. Use this information to derive the
Pythagorean theorem.

SECTION 8.1 Evaluating Roots 503

193.0 ft

110.0 ft

14.0 mi

19.0 mi Town B

Town A

x

(^58)
7
12
y
79.A surveyor wants to find the height of a
building. At a point 110.0 ft from the
base of the building, he sights to the top
of the building and finds the distance to
be 193.0 ft. How tall is the building?
80.Two towns are separated by dense
woods. To go from Town B to Town A, it
is necessary to travel due west for 19.0 mi
and then turn due north and travel for
14.0 mi. How far apart are the towns?
81.What is the value of x(to the nearest
thousandth) in the figure?
82.What is the value of y(to the nearest
thousandth) in the figure?
a a
a a
b
b
b
c
c
cc
b
a
b
b
c
c
a
84.James A. Garfield, the twentieth president of the United States, provided
a proof of the Pythagorean theorem using the given figure. Supply the
required information in each of parts (a) – (c) in order to follow
his proof.
(a)Find the area of the trapezoid WXYZusing the formula for the area of
a trapezoid.
(b)Find the area of each of the right triangles PWX, PZY, and PXY.
(c)Since the sum of the areas of the three right triangles must equal the area of the
trapezoid, set the expression from part (a) equal to the sum of the three expressions
from part (b). Simplify the equation as much as possible.
Find the distance between each pair of points. Express the answer as a whole number or as a
square root. Do not use a calculator. See Example 8.

85. and 86. and 87. and


86. and 89. and 90. and


87. and 92. and a-

8

5

,

1

2

a b

2

5

,

3

2

a b

3

4

, -

1

3

a- b

1

4

,

2

3

b

1 4, 6 2 1 - 4, - 92 1 - 1, - 22 1 - 3, 1 2 1 - 3, - 62 1 - 4, 0 2

1 5, 7 2 1 1, 4 2 1 8, 13 2 1 3, 1 2 1 2, 9 2 1 - 3, - 32

b

a

a

b

W

P

ZY

X

c

c