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a=____ b=____ c=____

Pythagorean Theorem. c 2. a=____ b=____ c=____. a 2. b 2. In any right triangle, the sum of the square of the lengths of the legs is equal to the square length of the hypotenuse. 1. The Pythagorean Theorem. 11.2. LESSON.

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a=____ b=____ c=____

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  1. Pythagorean Theorem c2 a=____ b=____ c=____ a2 b2 In any right triangle, the sum of the square of the lengths of the legs is equal to the square length of the hypotenuse 1

  2. The Pythagorean Theorem 11.2 LESSON 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?

  3. The Pythagorean Theorem 11.2 LESSON 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.

  4. The Pythagorean Theorem 11.2 LESSON Pythagorean Theorem Words For any right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. Algebra a2 + b2 = c2

  5. The Pythagorean Theorem 11.2 LESSON 1 EXAMPLE 273,428 = c 523 c 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. Pythagorean theorem a2 + b2 = c2 Substitute 212 for a and 478 for b. 2122 + 4782 = c2 Evaluate powers. 44,944 + 228,484 = c2 Add. 273,428 =c2 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.

  6. The Pythagorean Theorem 11.2 LESSON 2 EXAMPLE a = 44 a = 2 11 The unknown length a is 2 11 units. Finding the Length of a Leg Find the unknown length a in simplest form. Pythagorean theorem a2 +b2 = c2 Substitute. a2 + 102 = 122 Evaluate powers. a2 + 100 = 144 Subtract100from each side. a2 =44 Take positive square root of each side. Simplify. ANSWER

  7. The Pythagorean Theorem 11.2 LESSON Converse of the Pythagorean Theorem The Pythagorean theorem can be written in “if-then” form. Theorem: If a triangle is a right triangle, thena2 + b2 = c2. If you reverse the two parts of the statement, the new statement is called the converse of the Pythagorean theorem. Converse: Ifa2 + b2 = c2, 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.

  8. The Pythagorean Theorem 11.2 LESSON 3 EXAMPLE ? 32 + 52 = 72 ? 9 + 25 = 49 34 =49 Identifying Right Triangles Determine whether the triangle with the given side lengths is a right triangle. a = 3, b = 5, c = 7 SOLUTION a2 + b2 = c2 ANSWER Not a right triangle.

  9. The Pythagorean Theorem 11.2 LESSON 3 EXAMPLE ? 32 + 52 = 72 ? 9 + 25 = 49 ? 152 + 82 = 172 ? 225 + 64 = 289 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 a2 + b2 = c2 a2 + b2 = c2 34 =49 289 =289 ANSWER ANSWER Not a right triangle. A right triangle.

  10. c2 = a2 + b2 Use the Pythagorean Theorem. c2 = 282 + 212 Replace a with 28, and b with 21. Simplify. c2 = 1,225 Find the positive square root of each side. c = 1,225 = 35 The Pythagorean Theorem Lesson 11-2 Additional Examples Find c, the length of the hypotenuse. The length of the hypotenuse is 35 cm. Quick Check 11-2

  11. a2 + b2 =c2 Use the Pythagorean Theorem. 72 + x2 =142 Replace a with 7, b with x, and c with 14. 49 + x2 =196 Simplify. x2 =147 Subtract 49 from each side. x=147 Find the positive square root of each side. The Pythagorean Theorem Lesson 11-2 Additional Examples Find the value of x in the triangle. Round to the nearest tenth. 11-2

  12. Then use one of the two methods below to approximate . 147 Method 1: Use a calculator. A calculator value for Round to the nearest tenth. is 12.124356. Method 2: Use a table of square roots. Use the table on page 800. Find the number closest to 147 in the N2 column. Then find the corresponding value in the N column. It is a little over 12. x 12.1  147 x 12.1  Estimate the nearest tenth. The Pythagorean Theorem Lesson 11-2 Additional Examples (continued) Quick Check The value of x is about 12.1 in. 11-2

  13. c2 = a2 + b2 Use the Pythagorean Theorem. c2 = 102 + 102 Replace a with 10 (half the span), and b with 10. c2 = 100 + 100 Square 10. c2 = 200 Add. c = 200 Find the positive square root. c 14.1 Round to the nearest tenth. Lesson 11-2 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? Quick Check The rafter length should be about 14.1 ft. 11-2

  14. a2 + b2 = Write the equation for the Pythagorean Theorem. Replace a and b with the shorter lengths and c with the longest length. 102 + 242262 Simplify. 100 + 576 676 676 = 676 The Pythagorean Theorem Lesson 11-2 Is a triangle with sides 10 cm, 24 cm, and 26 cm a right triangle? c2 The triangle is a right triangle. Quick Check 11-2

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