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Understanding the Pythagorean Theorem: Right Triangles and Their Properties

This lesson explores the Pythagorean Theorem, a fundamental concept in geometry that applies to right triangles. It states that for any right triangle, the square of the length of the hypotenuse (C) is equal to the sum of the squares of the other two sides (A and B), expressed as A² + B² = C². Through examples and practice problems, students will learn how to identify right triangles, calculate missing side lengths, and determine triangle types (acute, right, obtuse) based on side lengths. Hands-on activities with graph paper reinforce the theorem's applicability.

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Understanding the Pythagorean Theorem: Right Triangles and Their Properties

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  1. The Pythagorean Theorem Lesson 7-1 Lesson 7-1: The Pythagorean Theorem

  2. The Pythagorean Theorem Given any right triangle, A2 + B2 = C2 C A B Lesson 7-1: The Pythagorean Theorem

  3. Example In the following figure if A = 3 and B = 4, Find C. A2 + B2 = C2 32 + 42 = C 2 9 + 16 = C2 5 = C C A B Lesson 7-1: The Pythagorean Theorem

  4. You can verify the Pythagorean Theorem with the following: Given a piece of graph paper, make a right triangle. Then make squares of the right triangle. Then find the square’s areas. Lesson 7-1: The Pythagorean Theorem

  5. A=8, B= 15, Find C A=7, B= 24, Find C A=9, B= 40, Find C A=10, B=24, Find C A =6, B=8, Find C Pythagorean Theorem : Examples C = 17 C = 25 C A C = 41 C = 26 B C = 10 Lesson 7-1: The Pythagorean Theorem

  6. Finding the legs of a right triangle: In the following figure if B = 5 and C = 13, Find A. A2 + B2 = C2 A2 +52 = 132 A2 + 25 = 169 A2+25-25=169-25 A2 = 144 A = 12 C A B Lesson 7-1: The Pythagorean Theorem

  7. 1) A=8, C =10 , Find B 2) A=15, C=17 , Find B 3) B =10, C=26 , Find A 4) A=15, B=20, Find C 5) A =12, C=16, Find B 6) B =5, C=10, Find A 7) A =6, B =8, Find C 8) A=11, C=21, Find B More Examples: B = 6 B = 8 A = 24 C = 25 C B = 10.6 A A = 8.7 C = 10 B = 17.9 B Lesson 7-1: The Pythagorean Theorem

  8. Given the lengths of three sides, how do you know if you have a right triangle? Given A = 6, B=8, and C=10, describe the triangle. A2 + B2 = C2 62 +82 = 102 36 + 64 = 100 This is true, so you have a right triangle. C A B Lesson 7-1: The Pythagorean Theorem

  9. Given A = 4, B = 5, and C =6, describe the triangle. A2 + B2 = C2 42 + 52 = 62 16 + 25 = 36 41 > 36, so we have an acute triangle. If A2 + B2 > C2, you have an acute triangle. A B C Lesson 7-1: The Pythagorean Theorem

  10. Given A = 4, B = 6, and C =8, describe the triangle. A2 + B2 = C2 42 + 62 = 82 16 + 36 = 64 52 < 64, so we have an obtuse triangle. If A2 + B2 < C2, you have an obtuse triangle. A B C Lesson 7-1: The Pythagorean Theorem

  11. 1) A=9, B=40, C=41 2) A=10, B=15, C=20 3) A=2, B=5, C=6 4) A=12, B=16, C=20 5) A=11, B=12, C=14 6) A=2, B=3, C=4 7) A=1, B=7, C=7 8) A=90, B=120, C=150 Describe the following triangles as acute, right, or obtuse right right obtuse right C acute A obtuse acute right B Lesson 7-1: The Pythagorean Theorem

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