Triangle ABC is an isosceles triangle Triangle ABC is an isosceles triangle. The angle measures of triangle ABC are in the ratio 2:5:2. What are the measures of the angles? A. 50°, 80°, 50° B. 40°, 100°, 40° C. 30°, 120°, 30° D. 20°, 140°, 20° A B C D 5-Minute Check 6

Splash Screen

You have already found missing measures of similar triangles You have already found missing measures of similar triangles. (Lesson 6–7) Use the Pythagorean Theorem to find the length of a side of a right triangle. Use the converse of the Pythagorean Theorem to determine whether a triangle is a right triangle. Then/Now

solving a right triangle converse legs The sides that form a right angle in a right triangle hypotenuse Pythagorean Theorem solving a right triangle converse The side opposite the right angle of a right triangle (the longest side) If a triangle is a right triangle, then the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs or c2 = a2 + b2 Using the Pythagorean Theorem to find the length of the third side of a right triangle, if you know the other two sides The statement formed by reversing the phrases after if and then in an if-then statement Vocabulary

Concept

Find the length of the hypotenuse of the right triangle. Find the Hypotenuse Length Find the length of the hypotenuse of the right triangle. c2 = a2 + b2 Pythagorean Theorem c2 = 212 + 202 Replace a with 21 and b with 20. c2 = 441 + 400 Evaluate 212 and 202. c2 = 841 Add 441 and 400. Definition of square root Use the principal square root. Answer: The length of the hypotenuse is 29 feet. Example 1

A B C D Find the length of the hypotenuse of the right triangle. A. 25 m B. 12.5 m C. 5 m D. 2.6 m A B C D Example 1

A ladder positioned against a 10-foot building reaches its top A ladder positioned against a 10-foot building reaches its top. Its base is 3 feet from the building. About how long is the ladder in feet? Round to the nearest tenth. Read the Test Item Make a drawing to illustrate the problem. The ladder, ground, and side of the house form a right triangle. Solve the Test Item Use the Pythagorean Theorem to find the length of the ladder. Example 2

c2 = a2 + b2 Pythagorean Theorem c2 = 32 + 102 Replace a with 3 and b with 10. c2 = 9 + 100 Evaluate 32 and 102. c2 = 109 Simplify. Definition of square root Use the principal square root. Answer: The ladder is about 10.4 feet tall. Example 2

An 18-foot ladder is placed against a building which is 14 feet tall An 18-foot ladder is placed against a building which is 14 feet tall. About how far is the base of the ladder from the building? A. 11.6 feet B. 10.9 feet C. 11.3 feet D. 11.1 feet A B C D Example 2

c2 = a2 + b2 Pythagorean Theorem Solve a Right Triangle LANDSCAPING A diagonal path through a rectangular garden is 32 feet long. The length of the garden is 24 feet. About how many feet wide is the garden? The diagonal is the hypotenuse of a right triangle. The length and width are the sides. c2 = a2 + b2 Pythagorean Theorem 322 = 242 + b2 Replace c with 32 and a with 24. 1024 = 576 + b2 Evaluate 322 and 242. 448 = b2 Subtract 576 from each side. Example 3

Definition of square root. Solve a Right Triangle Definition of square root. ENTER 2nd 448 21.16601049 Use a calculator. Answer: The garden is about 21.2 feet wide. Example 3

LANDSCAPING A diagonal path through a rectangular garden is 40 feet long. The length of the garden is 30 feet long. About how many feet wide is the garden? A. 26.5 feet B. 35 feet C. 50 feet D. 61.2 feet A B C D Example 3

c2 = a2 + b2 Pythagorean Theorem Identify a Right Triangle A. The measures of three sides of a triangle are given. Determine whether the triangle is a right triangle. 48 ft, 60 ft, 78 ft c2 = a2 + b2 Pythagorean Theorem ? 782 = 482 + 602 Replace c with 78, a with 48, and b with 60. 6084 = 2304 + 3600 Evaluate 782, 482, and 602. ? 6084 ≠ 5904 Simplify. Answer: The triangle is not a right triangle. Example 4 A

c2 = a2 + b2 Pythagorean Theorem Identify a Right Triangle B. The measures of three sides of a triangle are given. Determine whether the triangle is a right triangle. 24 cm, 70 cm, 74 cm c2 = a2 + b2 Pythagorean Theorem ? 742 = 242 + 702 Replace c with 74, a with 24, and b with 70. 5476 = 576 + 4900 Evaluate 742, 242, and 702. ? 5476 = 5476 Simplify. Answer: The triangle is a right triangle. Example 4

A. The measures of three sides of a triangle are given A. The measures of three sides of a triangle are given. Determine whether the triangle is a right triangle. 42 in., 61 in., 84 in. A. Yes, the triangle is a right triangle. B. No, the triangle is not a right triangle. A B Example 4 CYP A

B. The measures of three sides of a triangle are given B. The measures of three sides of a triangle are given. Determine whether the triangle is a right triangle. 16 m, 30 m, 34 m A. Yes, the triangle is a right triangle. B. No, the triangle is not a right triangle. A B Example 4 CYP B

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