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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
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
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 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.