Intermediate Algebra (11th edition)

In Section 9.1we will see that an equation such as has two solutions:

(the principal, or positive, square root of 7) and Similarly, has two

solutions, In applications we often choose only the principal

square root.

Using the Pythagorean Theorem

Use the Pythagorean theorem to find the length of the

unknown side of the triangle in FIGURE 7.

Pythagorean theorem Let and. Apply the exponents. Add. Interchange sides.

Choose the principal root.

Factor.

Product rule

Simplify.

The length of the hypotenuse is 2213. NOW TRY

c= 2213

c= 24 # 213

c= 24 # 13

c= 252

c^2 = 52

16 + 36 = c^2

42 + 62 = c^2 a= 4 b= 6

a^2 + b^2 = c^2

EXAMPLE 8

252 = 2213.

- 27. c^2 = 52

x^2 = 7 27

448 CHAPTER 8 Roots, Radicals, and Root Functions

Pythagorean Theorem

If aand bare the lengths of the shorter sides of a right triangle and cis the length

of the longest side, then

The two shorter sides are the legsof the triangle, and the longest side is the

hypotenuse.The hypotenuse is the side opposite the right angle.

a^2 b^2 c^2. a

c

Hypotenuse

Legs b

90 °

4

6

c

90 °

FIGURE 7

OBJECTIVE 6 Use the distance formula. The distance formulaallows us to

find the distance between two points in the coordinate plane, or the length of the line

segment joining those two points.

FIGURE 8on the next page shows the points and The vertical line

through and the horizontal line through intersect at the point

. Thus, the point becomes the vertex of the right angle in a right

triangle.

1 - 5, - 42 1 - 5, - 42

1 - 5, 3 2 1 3, - 42

1 3, - 42 1 - 5, 3 2.

Substitute carefully.

NOW TRY
EXERCISE 8
Find the length of the
unknown side in each triangle.

(a)

(b)

NOW TRY ANSWERS

(a) 289 (b) 623

8

90

c

5

6

b

12 90

CAUTION When substituting in the equation of the

Pythagorean theorem, be sure that the length of the hypotenuse is substituted for c

and that the lengths of the legs are substituted for aand b.

a^2 +b^2 =c^2 ,

OBJECTIVE 5 Use the Pythagorean theorem.The Pythagorean theorem

provides an equation that relates the lengths of the three sides of a right triangle.