1. ALGEBRA READINESS LESSON 3-6 Warm Up Lesson 3-6 Warm Up

2. ALGEBRA READINESS LESSON 3-5 Warm Up Lesson 3-4 Warm Up

3. ALGEBRA READINESS “The Pythagorean Theorem” (3-6) 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.

4. ALGEBRA READINESS Find the length of the hypotenuse of the triangle at the right. The length of the hypotenuse is 10 ft. a 2 + b 2 = c 2 Use the Pythagorean Theorem. 6 2 + 8 2 = c 2 Substitute 6 for a and 8 for b. 36 + 64 = c 2 Square 6 and 8. 100 = c 2 Add. 10 = cSimplify. 100 = c 2 Find the positive square root of each side. Pythagorean Theorem LESSON 3-6 Additional Examples

5. ALGEBRA READINESS A wheelchair ramp leads into an apartment building doorway that is 5 feet above the ground. The horizontal distance from the entrance to the end of the ramp is 16 feet. What is the length in feet of the ramp? a 2 + b 2 = c 2 Use the Pythagorean Theorem. 5 2 + 16 2 = c 2 Substitute 5 for a and 16 for b. 25 + 256 = c 2 Square 5 and 16. 281 = c 2 Add. 281 = c 2 Find the positive square root of each side. 281 = cSimplify. The length of the ramp is 281 or about 16.8 feet. Pythagorean Theorem LESSON 3-6 Additional Examples

6. ALGEBRA READINESS A toy fire truck is positioned so that the base of the ladder is 13 cm from the wall. The ladder is extended 28 cm to the wall. How high above the table is the top of the ladder? Define:Let b = height (in cm) of the ladder from a point 9 cm above the table. Words: The triangle formed is a right triangle. Use the Pythagorean Theorem. The Pythagorean Theorem LESSON 3-6 Additional Examples

7. ALGEBRA READINESS Translate: a 2 + b 2 = c 2 13 2 + b 2 = 28 2 Substitute. 169 + b 2 = 784Simplify. b 2 = 615Subtract 169 from each side. b 2 = 615Find the principal square root of each side. b 24.8Use a calculator and round to the nearest tenth. (continued) The height to the top of the ladder is 9 cm higher than 24.8 cm, so it is about 33.8 cm from the table. The Pythagorean Theorem LESSON 3-6 Additional Examples

8. ALGEBRA READINESS “The Pythagorean Theorem” (3-6) What is the “Converse (Opposite) 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. a 2 + b 2 = c 2 Make a and b the two a 2 + b 2 = c 2 shorter sides. Use the Converse of the Pythagorean Theorem

9. ALGEBRA READINESS Determine whether the given lengths can be sides of a right triangle. a. 5 in., 5 in., and 7 in. This triangle is not a right triangle. b. 10 cm, 24 cm, and 26 cm This triangle is a right triangle. 100 + 576 676Simplify. 676 = 676 10 2 + 24 2 26 2 Determine whether a 2 + b 2 = c 2, where c is the longest side. 5 2 + 5 2 7 2 Determine whether a 2 + b 2 = c 2, where c is the longest side. 25 + 25 49Simplify. 50 = 49 / The Pythagorean Theorem LESSON 3-6 Additional Examples

10. ALGEBRA READINESS Find the missing side length of each right triangle, where a and b represent the lengths of the two legs. 1.a = ?, b = 24, c = 30 2.a = 8, b = ?, c = 10 3.a = 5, b = 3, c = ? 18 6 √34 or about 5.8 Pythagorean Theorem LESSON 3-6 Lesson Quiz