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Chapter 7 Lesson 2

Chapter 7 Lesson 2. Objective: To use the Pythagorean Theorem and its converse. Theorem 7-4   Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a 2 + b 2 = c 2.

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Chapter 7 Lesson 2

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  1. Chapter 7 Lesson 2 Objective: To use the Pythagorean Theorem and its converse.

  2. Theorem 7-4   Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a2 + b2 = c2

  3. A Pythagorean triple is a set of nonzero whole numbers a, b, and c that satisfy the equation a2 + b2 = c2. Here are some common Pythagorean triples. 3,4,5 5,12,12 8,15,17 7,24,25

  4. Example 1: Pythagorean Triples Find the length of the hypotenuse of ∆ABC. Do the lengths of the sides of ∆ABC form a Pythagorean triple? The lengths of the sides, 20, 21, and 29, form a Pythagorean triple.

  5. Example 2: Pythagorean Triple A right triangle has a hypotenuse of length 25 and a leg of length 10. Find the length of the other leg. Do the lengths of the sides form a Pythagorean triple? Not a Pythagorean Triple

  6. Example 3: Using Simplest Radical Form Find the value of x. Leave your answer in simplest radical form.     

  7. Example 4: Using Simplest Radical Form The hypotenuse of a right triangle has length 12. One leg has length 6. Find the length of the other leg. Leave the answer in simplest radical form.

  8. 12 m h 12 m 12 m 10 m h 20 m Example 5: Find the area of the triangle. 102+h2=122 100+h2=144 h2=44 h=√44 h=2√11 A=(1/2)bh A=(1/2)(20)(2√11) A=20√11

  9. 7 cm √53 cm Example 6: Find the area of the triangle. 72+b2=(√53)2 49+b2=53 b2=4 b=2 A=(1/2)bh A=(1/2)(7)(2) A=7

  10. Theorem 7-5:Converse of the Pythagorean Theorem If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

  11. 85 13 84 Example 7: Is this triangle a right triangle? c2=a2+b2 852=132+842 7225=169+7056 7225=7225

  12. Example 8: A triangle has sides of lengths 16, 48, and 50. Is the triangle a right triangle? c2=a2+b2 502=162+482 2500=256+2304 2500≠2560 Not a right triangle.

  13. Theorem 7-6: If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, the triangle is obtuse. If c2>a2+b2, the triangle is obtuse. c a b

  14. Theorem 7-7:If the squares of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, the triangle is acute. If c2<a2+b2, the triangle is acute. b a c

  15. Example 9: The lengths of the sides of a triangle are given. Classify each triangle as acute, obtuse, or right. b.) 12,13,15 152=132+122 225=169+144 225<313 ACUTE a.) 6,11,14 142=62+112 196=36+121 196>157 OBTUSE

  16. Assignment: Page 361 – 363 #1-43

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