Basic Mathematics for College Students

748 Chapter 9 An Introduction to Geometry

SolutionWe will let and , and substitute into the Pythagorean equation

to find.

This is the Pythagorean equation.

Substitute 3 for aand 4 for b.

Evaluate each exponential expression.

Do the addition.

Reverse the sides of the equation so that c^2 is on the left.

To find ,we must find a number that, when squared, is 25. There are two such

numbers, one positive and one negative; they are the square roots of 25. Since

represents the length of a side of a triangle, cannot be negative. For this reason,

we need only find the positive square root of 25 to get.

The symbol is used to indicate the positive square root of a number.

because 5^2 25.

The length of the hypotenuse is 5 in.

c 5 125  5

c 125 2

c

c

c

c

c^2  25

25 c^2

9  16 c^2

32  42 c^2

a^2 b^2 c^2

c

a 3 b 4

a = 3 in. c

b = 4 in.

Success Tip The Pythagorean theorem is used to find the lengths of sides of

right triangles. A calculator with a square root key is often helpful in the

final step of the solution process when we must find the positive square root of

a number.

1

Self Check 2

In Example 2, can the crews

communicate by radio if the

distance from point to point

remains the same but the

distance from point to point

increases to 2,520 yards?

Now TryProblems 19 and 43

A C

B C

Now TryProblem 15

1,000 yd

A

C 2,400 yd B

c

EXAMPLE (^2) Firefighting To fight a forest fire, the forestry department
plans to clear a rectangular fire break around the fire, as shown in the following
figure. Crews are equipped with mobile communications that have a 3,000-yard
range. Can crews at points and remain in radio contact?
StrategyWe will use the Pythagorean theorem to find the distance between
points and.
WHYIf the distance is less than 3,000 yards, the crews can communicate by radio.
If it is greater than 3,000 yards, they cannot.
SolutionThe line segments connecting
points ,,and form a right triangle.
To find the distance from point to
point , we can use the Pythagorean
equation, substituting 2,400 for and
1,000 for and solving for.
This is the Pythagorean equation.
Substitute for aand b.
Evaluate each exponential expression.
Do the addition.
Reverse the sides of the equation so that
c^2 is on the left.
If c^2 6,760,000, then cmust be a
square root of 6,760,000. Because c
represents a length, it must be the positive
square root of 6,760,000.
Use a calculator to find the square root.
The two crews are 2,600 yards apart. Because this distance is less than the 3,000-
yard range of the radios, they can communicate by radio.
c2,600
c 1 6,760,000
c^2 6,760,000
6,760,000c^2
5,760,0001,000,000c^2
2,400^2 1,000^2 c^2
a^2 b^2 c^2
b c
a

B

c A

AB C

A B

A B