1. PRE-ALGEBRA

2. Lesson 11-2 Warm-Up

3. PRE-ALGEBRA What are the parts of a right triangle? What is the Pythagorean Theorem? hypotenuse: The side opposite of the right angle (the longest side of a right triangle) legs: the sides that form the right angle (the shortest sides of the triangle) Pythagorean Theorem: In any right triangle, the sum of the squares of the length of the legs, a and b, is equal to the square of the length of the hypotenuse, c. Note: The legs terms, “a” and “b” are interchangeable (can be switched) since you’re adding them together (Commutative Property), but the hypotenuse is always the “c” term. The Pythagorean Theorem (11-2)

4. PRE-ALGEBRA Find c, the length of the hypotenuse. 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 = 28 2 + 21 2 LESSON 11-2 Additional Examples The Pythagorean Theorem

5. PRE-ALGEBRA Find the value of x in the triangle. Round to the nearest tenth. x = 147 x 2 = 147 Find the positive square root of each side. Subtract 49 from each side. a 2 + b 2 = c 2 49 + x 2 = 196 7 2 + x 2 = 14 2 Use the Pythagorean Theorem. Simplify. Replace a with 7, b with x, and c with 14. LESSON 11-2 Additional Examples The Pythagorean Theorem

6. PRE-ALGEBRA Then, use one of the two methods below to approximate. 147 (continued) The value of x is about 12.1 in. Estimate the nearest tenth.x 12.1 Use the table on page 701. 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 estimation. The number 147 lies between the perfect squares 144 and 169. Since 147 is closer to 144 than to 169, is closer to 12. So, ≈ 12. 147 LESSON 11-2 Additional Examples The Pythagorean Theorem

7. PRE-ALGEBRA 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? 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 the 10s..c 2 = 100 + 100 Replace a with 10 (half the span), and b with 10.c 2 = 10 2 + 10 2 LESSON 11-2 Additional Examples The Pythagorean Theorem

8. PRE-ALGEBRA “The Pythagorean Theorem” (10-2) What is the “Converse (Opposite) of the Pythagorean Theorem”? How can you use the Converse of the Pythagorean Theorem? Converse of the Pythagorean Theorem: If a triangle has sides a, b, and c, and a 2 + b 2 = c 2 (in other words, if the sum of the squares of the shorter sides equal the square of the longest side), then the triangle is a right triangle with hypotenuse c. The Converse (Opposite) of the Pythagorean Theorem can be use to determine whether or not a triangle is a right triangle. Example: Determine whether the given lengths can form a right triangle. 5 in., 12 in., and 13 in. 7 in., 9 in., and 12 in. Make a and b the two shorter sides. Use the Converse of the Pythagorean Theorem. The triangle is a right triangle. The triangle is not a right triangle.

9. PRE-ALGEBRA Is a triangle with sides 10 cm, 24 cm, and 26 cm a right triangle? Yes, the triangle is a right triangle. Simplify.100 + 576 676 Replace a and b with the shorter lengths and c with the longest length. 10 2 + 24 2 26 2 a 2 + b 2 = c 2 Write the equation for the Pythagorean Theorem. 676 = 676 LESSON 11-2 Additional Examples The Pythagorean Theorem

10. PRE-ALGEBRA Find the missing length. Round to the nearest tenth. 1.a = 7, b = 8, c = 2.a = 9, c = 17, b = 3.Is a triangle with sides 6.9 ft, 9.2 ft, and 11.5 ft a right triangle? Explain. 4.What is the rise of a roof if the span is 30 ft and the rafter length is 16 ft? Refer to the diagram on page 572. about 5.6 ft 10.6 14.4 yes; 6.9 2 + 9.2 2 = 11.5 2 LESSON 11-2 The Pythagorean Theorem Lesson Quiz