Warm-Up Exercises

2. Solve x2 + 9 = 25.

ANSWER 10, –10

ANSWER 4, –4

1. Solve x2 = 100.

ANSWER 2 5

3. Simplify 20.

Warm-Up Exercises

ANSWER 6 cm

4. Find x.

Warm-Up ExercisesEXAMPLE 1 Find the length of a hypotenuse

SOLUTION

Find the length of the hypotenuse of the right triangle.

(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.

Warm-Up ExercisesGUIDED PRACTICE for Example 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.

1.

ANSWER Leg; 4

Warm-Up ExercisesGUIDED PRACTICE for Example 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.

2.

hypotenuse; 2 13ANSWER

Warm-Up ExercisesEXAMPLE 2 Standardized Test Practice

SOLUTION

= +

Warm-Up ExercisesEXAMPLE 2 Standardized Test Practice

Find positive square root.

Substitute.

Multiply.

Subtract 16 from each side.

SOLUTION

Approximate with a calculator.

162 = 42 + x2

256 = 16 + x2

15.492 ≈ x

240 = x

240 = x2

ANSWER

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

The correct answer is D.

Warm-Up ExercisesGUIDED PRACTICE for Example 2

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

Warm-Up ExercisesGUIDED PRACTICE for Example 2

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

4.

right triangleANSWER

Warm-Up ExercisesEXAMPLE 3Find the area of an isosceles triangle

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.

Warm-Up ExercisesEXAMPLE 3Find the area of an isosceles triangle

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

Warm-Up ExercisesEXAMPLE 3Find the area of an isosceles triangle

Find the area.

STEP 3

= (10) (12) = 60 m212

ANSWER

The area of the triangle is 60 square meters.

Area = 12

(base) (height)

Warm-Up ExercisesGUIDED PRACTICE for Example 3

5. Find the area of the triangle.

ANSWER about 149.2 ft2.

Warm-Up ExercisesGUIDED PRACTICE for Example 3

Find the area of the triangle.6.

ANSWER 240 m2.

Warm-Up ExercisesEXAMPLE 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 a hypotenuse using two methods

Find the length of the hypotenuse of the right triangle.

Warm-Up ExercisesEXAMPLE 4 Find the length of a hypotenuse using two methods

SOLUTION

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.

Warm-Up ExercisesGUIDED PRACTICE for Example 4

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

7.

ANSWER 15 in.

8.

ANSWER 50 cm.

Warm-Up ExercisesDaily Homework Quiz

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

ANSWER 39

Warm-Up ExercisesDaily Homework Quiz

2. Find the area of the isosceles triangle.

ANSWER 1080 cm2

Warm-Up ExercisesDaily Homework Quiz

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

ANSWER 4 13