Beginning Algebra, 11th Edition

Using the Pythagorean Theorem to Solve an Application

A ladder 10 ft long leans against a wall. The foot of the ladder is 6 ft from the base of

the wall. How high up the wall does the top of the ladder rest?

Step 1 Readthe problem again.

Step 2 Assign a variable.As shown in FIGURE 2, a right

triangle is formed with the ladder as the hypotenuse.

Let arepresent the height of the top of the ladder,

measured straight down to the ground.

Step 3 Write an equationusing the Pythagorean theorem.

a^2 + 62 = 102 Let and c= 10 b= 6.

a^2 +b^2 =c^2

EXAMPLE 7

498 CHAPTER 8 Roots and Radicals

NOW TRY
EXERCISE 7
The length of a rectangle is
48 ft, and the width is 14 ft.
Find the measure of the
diagonal of the rectangle.

a

6 ft

10 ft

FIGURE 2

14 ft

48 ft

Diagonal

Step 4 Solve. Square.

Subtract 36.

Solve for a.

We choose the positive square root of 64 because arepresents a length.

Step 5 State the answer.The top of the ladder rests 8 ft up the wall.

Step 6 Check.From FIGURE 2, we must have the following.

✓ True

The check confirms that the top of the ladder rests 8 ft up the wall.

NOW TRY

64 + 36 = 100

82 + 62  102 a^2 +b^2 =c^2

a= 8 264 = 8

a= 264

a^2 = 64

a^2 + 36 = 100

Substitute

carefully.

Although Pythagoras may have written the first proof of the Pythagorean relation-

ship, there is evidence that the Babylonians knew the concept quite well. The figure

illustrates the theorem with a tile pattern. The side of the square along the hypotenuse

measures 5 units, while the sides along the legs measure 3 and 4 units. If we let

and the Pythagorean theorem is satisfied.

✓ True

For Discussion or Writing

The diagram to the right can be used to verify the Pythagorean

theorem, as follows.

1.Find the area of the large square.

2.Find the sum of the areas of the smaller square and the four

right triangles.

3.Set the areas equal and simplify the equation.

25 = 25

32 + 42  52

a^2 +b^2 =c^2

a=3,b=4, c=5,

CONNECTIONS

a

b

ab

a

b

b a

c

c

c

c

NOW TRY ANSWER
7.50 ft

(Note that the diagonal di-
vides the rectangle into two
right triangles with itself as
the hypotenuse.)

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