Proving Right Triangles Pythagorean Theorem Proving Right Triangles

The Pythagorean Theorem Converse of the Pythagorean Theorem The Pythagorean theorem can be written in “if-then” form. Theorem: If a triangle is a right triangle, then a 2 + b 2 = c 2. If you reverse the two parts of the statement, the new statement is called the converse of the Pythagorean theorem. Converse: If a 2 + b 2 = c 2, then the triangle is a right triangle. Although not all converses of true statements are true, the converse of the Pythagorean theorem is true. You can use it to determine whether a triangle is a right triangle.

The Pythagorean Theorem EXAMPLE 3 Identifying Right Triangles Determine whether the triangle with the given side lengths is a right triangle. a = 3, b = 5, c = 7 SOLUTION a 2 + b 2 = c 2 3 2 + 5 2 = 7 2 ? 9 + 25 = 49 ? 34 ≠ 49 ANSWER Not a right triangle.

The Pythagorean Theorem EXAMPLE 3 Identifying Right Triangles Determine whether the triangle with the given side lengths is a right triangle. a = 3, b = 5, c = 7 a = 15, b = 8, c = 17 SOLUTION SOLUTION a 2 + b 2 = c 2 a 2 + b 2 = c 2 3 2 + 5 2 = 7 2 ? 15 2 + 8 2 = 17 2 ? 9 + 25 = 49 ? 225 + 64 = 289 ? 34 ≠ 49 289 = 289 ANSWER Not a right triangle. ANSWER A right triangle.

Is the following a right triangle? 4 ft 6 ft 3 ft

Is the following a right triangle? 8 cm 6 cm 10 cm

Is the following a right triangle? 7 cm 10 cm 9 cm

Example Which number completes the Pythagorean triple? 16, ____, 34 D. 30

Example Which number completes the Pythagorean triple? ____, 35, 37 D. 24

Example Which set of dimensions is a Pythagorean triple? A. 6 in., 8 in., 10 in. B. 10 in., 15 in., 18 in. C. 15 in., 25 in., 30 in. D. 18 in., 24 in., 27 in.

Example Jacob constructed a right triangle using drinking straws. Which are possible straw lengths that formed the triangle? A. 3 cm, 4 cm, 6 cm B. 2 cm, 5 cm, 7 cm C. 6 cm, 8 cm, 12 cm D. 5 cm, 12 cm, 13 cm

Example Which are not the dimensions of a right triangle? A. 3 in, 4 in, 5 in B. 12 in, 16 in, 20 in C. 7 in, 10 in, 16 in D. 6 in, 8 in, 10 in

5. The measures of three sides of a triangle are given below 5. The measures of three sides of a triangle are given below. Determine whether each triangle is a right triangle. , 3, and 8 Which side is the biggest? The square root of 73 (= 8.5)! This must be the hypotenuse (c). Plug your information into the Pythagorean Theorem. It doesn’t matter which number is a or b.

Sides: , 3, and 8 32 + 82 = ( ) 2 9 + 64 = 73 73 = 73 Since this is true, the triangle is a right triangle!! If it was not true, it would not be a right triangle.

Determine whether the triangle is a right triangle given the sides 6, 9, and Yes No Purple