The Pythagorean Theorem and Its Converse

Objectives Use the Pythagorean Theorem Use the Converse of the Pythagorean Theorem

The Pythagorean Theorem Theorem 7.4 In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the legs. a2 + b2 = c2

Proof of Pythagorean Theorem REASONS STATEMENTS Draw a ┴ from C to AB and Perpendicular Postulate Geometric Mean Theorem Cross Product Property 1. c a c b = = 2. a e b f 3. ce = a2 and cf = b2 4. ce + cf = a2 + b2 4. Addition Property 5. c(e + f) = a2 + b2 5. Distributive Property 6. e + f = c 6. Segment Add. Postulate 7. c2 = a2 + b2 7. Substitution Property

Example 1: LONGITUDE AND LATITUDE Carson City, Nevada, is located at about 120 degrees longitude and 39 degrees latitude. NASA Ames is located about 122 degrees longitude and 37 degrees latitude. Use the lines of longitude and latitude to find the degree distance to the nearest tenth degree if you were to travel directly from NASA Ames to Carson City.

Example 1: The change in longitude between NASA Ames and Carson City is or 2 degrees. Let this distance be a. The change in latitude is or 2 degrees latitude. Let this distance be b. Use the Pythagorean Theorem to find the distance in degrees from NASA Ames to Carson City, represented by c.

Example 1: Pythagorean Theorem Simplify. Add. Take the square root of each side. Use a calculator. Answer: The degree distance between NASA Ames and Carson City is about 2.8 degrees.

Example 2: Find d.

Example 2: Pythagorean Theorem Simplify. Subtract 9 from each side. Take the square root of each side. Use a calculator. Answer:

Your Turn: Find x. Answer:

Converse of the Pythagorean Theorem Theorem 7.5 If the square of the length of the longest side of the triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. If c2 = a2 + b2, then ∆ABC is a right triangle.

Example 3: COORDINATE GEOMETRY Verify that is a right triangle.

Example 3: Use the Distance Formula to determine the lengths of the sides. Subtract. Simplify. Subtract. Simplify.

Example 3: Subtract. Simplify. By the 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.

Example 3: Converse of the Pythagorean Theorem Simplify. Add. Answer: Since the sum of the squares of two sides equals the square of the longest side, is a right triangle.

Your Turn: COORDINATE GEOMETRY Verify that is a right triangle. Answer: is a right triangle because

Pythagorean Triples A Pythagorean Triple is three whole numbers that satisfy the equation a2 + b2 = c2 , where the longest side is the hypotenuse, c, and a and b are the two legs. i.e. 3, 4, 5

Example 4a: 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.

Example 4a: 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 4b: 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 4c: Determine whether 4, and 8 are the sides of a right triangle. Then state whether they form a Pythagorean triple. Pythagorean Theorem Simplify. Add. 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.

Your Turn: 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. 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.