7.1

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Apply the Pythagorean Theorem

7.1 Warm-Up

2. Solve x2 + 9 = 25.

ANSWER 10, –10

ANSWER 4, –4

1. Solve x2 = 100.

ANSWER 2 5

3. Simplify 20.

7.1 Warm-Up

ANSWER 6 cm

4. Find x.

7.1

Find the length of the hypotenuse of the right triangle.

Example 1

SOLUTION

(hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem

x2 = 62 + 82

x2 = 36 + 64

x2 = 100

x = 10 Find the positive square root.

Substitute.

Multiply.

Add.

7.1 Guided Practice

Identify the unknown side as a leg or hypotenuse. Then, find the unknown side length of the right triangle. Write your answer in simplest radical form.

1.

ANSWER Leg; 4

7.1

Identify the unknown side as a leg or hypotenuse. Then, find the unknown side length of the right triangle. Write your answer in simplest radical form.

Guided Practice

2.

hypotenuse; 2 13ANSWER

7.1 Example 2

SOLUTION

= +

7.1 Example 2

Find positive square root.

Substitute.

Multiply.

Subtract 16 from each side.

Approximate with a calculator.

162 = 42 + x2

256 = 16 + x2

15.492 ≈ x

240 = x

240 = x2

The ladder is resting against the house at about 15.5 feet above the ground.

ANSWER The correct answer is D.

7.1 Guided Practice

The top of a ladder rests against a wall, 23 feet above the ground. The base of the ladder is 6 feet away from the wall. What is the length of the ladder?

3.

about 23.8 ftANSWER

7.1 Guided Practice

The Pythagorean Theorem is only true for what type of triangle?

4.

right triangleANSWER

7.1 Example 3

SOLUTION

Find the area of the isosceles triangle with side lengths 10 meters, 13 meters, and 13 meters.

STEP 1 Draw a sketch. By definition, the length of an altitude is the height of a triangle. In an isosceles triangle, the altitude to the base is also a perpendicular bisector. So, the altitude divides the triangle into two right triangles with the dimensions shown.

7.1 Example 3

Use the Pythagorean Theorem to find the height of the triangle.STEP 2

Pythagorean Theorem

Substitute.

Multiply.

Subtract 25 from each side.

Find the positive square root.

c2 = a2 + b2

12 = h

132 = 52 + h2

169 = 25 + h2

144 = h2

7.1 Example 3

Find the area.STEP 3

= (10) (12) = 60 m212

The area of the triangle is 60 square meters.

Area = 12

(base) (height)

7.1 Guided Practice

Find the area of the triangle.

5.

ANSWER about 149.2 ft2

7.1 Guided Practice

Find the area of the triangle.

6.

ANSWER 240 m2.

7.1 Example 4

SOLUTION

Method 1: Use a Pythagorean triple.

A common Pythagorean triple is 5, 12, 13. Notice that if you multiply the lengths of the legs of the Pythagorean triple by 2, you get the lengths of the legs of this triangle: 5 2 = 10 and 12 2 = 24. So, the length of the hypotenuse is 13 2 = 26.

Find the length of the hypotenuse of the right triangle.

7.1 Example 4

Method 2: Use the Pythagorean Theorem.

x2 = 102 + 242

x2 = 100 + 576

x2 = 676

x = 26

Pythagorean Theorem

Multiply.

Add.

Find the positive square root.

7.1 Guided Practice

7.

ANSWER 15 in.

Find the unknown side length of the right triangle using the Pythagorean Theorem. Then use a Pythagorean triple.

7.1 Guided Practice

Find the unknown side length of the right triangle using the Pythagorean Theorem. Then use a Pythagorean triple.

8.

ANSWER 50 cm.

7.1 Lesson Quiz

1. Find the length of the hypotenuse of the right triangle.

ANSWER 39

7.1 Lesson Quiz

2. Find the area of the isosceles triangle.

ANSWER 1080 cm2

7.1 Lesson Quiz

3. Find the unknown side length x. Write your answer in simplest radical form.

ANSWER 4 13