5-Minute Check on Lesson 7-1 Transparency 7-2 Click the mouse button or press the Space Bar to display the answers. Find the geometric mean between each pair of numbers. State exact answers and answers to the nearest tenth and √5 and 5√5 3. Find the altitude 4. Find x, y, and z 5. Which of the following is the best estimate of x? Standardized Test Practice: ACBD y x z 4 20 x √117 ≈ 10.8 x = 8, y = √80 ≈ 8.9 z = √320 ≈ 17.9 √24 ≈ 4.9 C √50 ≈ 7.1

Lesson 7-2 Pythagorean Theorem and its Converse

Objectives Use the Pythagorean Theorem –If a right triangle, then c² = a² + b² Use the converse of the Pythagorean Theorem –If c² = a² + b², then a right triangle

Vocabulary None new

Pythagorean Theorem a 2 + b 2 = c 2 Sum of the squares of the legs is equal to the square of the hypotenuse a b c Converse of the Pythagorean Theorem: If the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle Remember from our first computer quiz: In an acute triangle, c 2 < a 2 + b 2. In an obtuse triangle, c 2 > a 2 + b 2.

Example 1 Find d. Answer: Pythagorean Theorem Simplify. Subtract 9 from each side. Take the square root of each side. Use a calculator.

Example 2 Find x. Answer:

Pythagorean Triples For three numbers to be a Pythagorean triple they must satisfy both of the following conditions: –They must satisfy c 2 = a 2 + b 2 where c is the largest number –All three must be whole numbers (integers) Common Pythagorean Triples: 3, 4, 5 5, 12, 13 8, 15, 17 6, 8, 10 9, 12, 15 7, 24, 25 12, 16, 20 9, 40, 41 15, 20, 25 10, 24, 26 16, 30, 34

Example 3 Determine whether 9, 12, and 15 are the sides of a right triangle. Then state whether they form a Pythagorean triple. Since the measure of the longest side is 15, 15 must be c. Let a and b be 9 and 12. Pythagorean Theorem Simplify. Add. Answer: These segments form the sides of a right triangle since they satisfy the Pythagorean Theorem. The measures are whole numbers and form a Pythagorean triple.

Example 4 Determine whether 21, 42, and 54 are the sides of a right triangle. Then state whether they form a Pythagorean triple. Pythagorean Theorem Simplify. Add. Answer: Since, segments with these measures cannot form a right triangle. Therefore, they do not form a Pythagorean triple.

Example 5 Pythagorean Theorem Simplify. Add. Determine whether 4, and 8 are the sides of a right triangle. Then state whether they form a Pythagorean triple. Answer: Since 64 = 64, segments with these measures form a right triangle. However, is not a whole number. Therefore, they do not form a Pythagorean triple.

Example 6 Answer: The segments form the sides of a right triangle and the measures form a Pythagorean triple. Answer: The segments do not form the sides of a right triangle, and the measures do not form a Pythagorean triple. Answer: The segments form the sides of a right triangle, but the measures do not form a Pythagorean triple. Determine whether each set of measures are the sides of a right triangle. Then state whether they form a Pythagorean triple. a. 6, 8, 10 b. 5, 8, 9 c.

Summary & Homework Summary: –The Pythagorean Theorem can be used to find the measures of the sides of a right triangle –If the measures of the sides of a triangle form a Pythagorean triple, then the triangle is a right triangle Homework: –pg 354, 17-19, 22-25, 30-35