11 a=____ b=____ c=____ a2a2 c2c2 b2b2 Pythagorean Theorem In any right triangle, the sum of the square of the lengths of the legs is equal to the square length of the hypotenuse

2 Bridges The William H. Harsha Bridge is a cable- stayed bridge that spans the Ohio River between Maysville, Kentucky, and Aberdeen, Ohio. About how long is the cable shown in red? The Pythagorean Theorem 11.2 LESSON

3 In a right triangle, the hypotenuse is the side opposite the right angle. The legs are the sides that form the right angle. The lengths of the legs and the length of the hypotenuse of a right triangle are related by the Pythagorean theorem. The Pythagorean Theorem 11.2 LESSON

4 Pythagorean Theorem For any right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. Words Algebra a 2 + b 2 = c 2 The Pythagorean Theorem 11.2 LESSON

5 EXAMPLE 1 Finding the Length of a Hypotenuse To find the length (to the nearest foot) of the cable on the William H. Harsha bridge if the tower is 212 feet and bridge surface is 478 feet, use the right triangle formed by the tower, the bridge surface, and the cable. a 2 + b 2 = c 2 Pythagorean theorem Substitute 212 for a and 478 for b. 44, ,484 = c 2 Evaluate powers = c 2 Add. 273,428 = c 2 Take positive square root of each side. Use a calculator. Round to nearest whole number. ANSWER The length of the cable is about 523 feet. The Pythagorean Theorem 11.2 LESSON 273,428 = c 523 c

6 EXAMPLE 2 Finding the Length of a Leg Find the unknown length a in simplest form. a 2 + b 2 = c 2 Pythagorean theorem Substitute. a = 144 Evaluate powers. a = 12 2 Subtract 100 from each side. a 2 = 44 Take positive square root of each side. Simplify. ANSWER The Pythagorean Theorem 11.2 LESSON a = 44 a = 2 11 The unknown length a is 2 11 units.

7 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 11.2 LESSON

8 EXAMPLE 3 Identifying Right Triangles Determine whether the triangle with the given side lengths is a right triangle. a 2 + b 2 = c 2 a = 3, b = 5, c = = 7 2 ? = 49 ? SOLUTION ANSWER Not a right triangle. The Pythagorean Theorem 11.2 LESSON 34 = 49

9 EXAMPLE 3 Identifying Right Triangles Determine whether the triangle with the given side lengths is a right triangle. a 2 + b 2 = c 2 a = 3, b = 5, c = = 7 2 ? = 49 ? SOLUTION ANSWER Not a right triangle. a 2 + b 2 = c = 289 a = 15, b = 8, c = = 17 2 ? = 289 ? SOLUTION ANSWER A right triangle. The Pythagorean Theorem 11.2 LESSON 34 = 49

10 The Pythagorean Theorem Find c, the length of the hypotenuse. Lesson 11-2 c 2 = a 2 + b 2 Use the Pythagorean Theorem. c 2 = 1,225 Simplify. c = 1,225 = 35 Find the positive square root of each side. The length of the hypotenuse is 35 cm. Replace a with 28, and b with 21.c 2 = Quick Check Additional Examples 11-2

11 The Pythagorean Theorem Find the value of x in the triangle. Round to the nearest tenth. Lesson 11-2 x = 147 x 2 = 147 Find the positive square root of each side. Subtract 49 from each side. a 2 + b 2 = c x 2 = x 2 = 14 2 Use the Pythagorean Theorem. Simplify. Replace a with 7, b with x, and c with 14. Additional Examples 11-2

12 Then use one of the two methods below to approximate. 147 The Pythagorean Theorem (continued) Lesson 11-2 The value of x is about 12.1 in. Estimate the nearest tenth. x 12.1 Use the table on page 800. Find the number closest to 147 in the N 2 column. Then find the corresponding value in the N column. It is a little over 12. Method 2: Use a table of square roots. Method 1: Use a calculator. is A calculator value for 147 Round to the nearest tenth.x 12.1 Quick Check Additional Examples 11-2

13 The carpentry terms span, rise, and rafter length are illustrated in the diagram. A carpenter wants to make a roof that has a span of 20 ft and a rise of 10 ft. What should the rafter length be? Lesson 11-2 The rafter length should be about 14.1 ft. c 2 = a 2 + b 2 Use the Pythagorean Theorem. Round to the nearest tenth.c 14.1 Find the positive square root.c = 200 Add.c 2 = 200 Square 10.c 2 = Replace a with 10 (half the span), and b with 10.c 2 = Quick Check 11-2

14 The Pythagorean Theorem Is a triangle with sides 10 cm, 24 cm, and 26 cm a right triangle? Lesson 11-2 The triangle is a right triangle. Simplify Replace a and b with the shorter lengths and c with the longest length a 2 + b 2 =Write the equation for the Pythagorean Theorem. 676 = 676 Quick Check 11-2 c2c2