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Exploring the Pythagorean Theorem: Proofs, Examples, and Applications

Discover the Pythagorean Theorem, its algebraic proof, Pythagorean triples, and real-world applications. Dive into solving problems involving right triangles, finding missing leg lengths, and calculating areas. This comprehensive guide covers essential questions, activators, definitions, and practical examples to enhance your understanding of this fundamental concept in geometry.

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Exploring the Pythagorean Theorem: Proofs, Examples, and Applications

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  1. The Pythagorean Theorem

  2. Essential Questions • How do I prove the Pythagorean Theorem? • How do I use the Pythagorean Theorem to solve problems?

  3. Activator • Solve. Round to the nearest tenth. • c2 = 36 • c2 = 84.2 • c2 = 28.2 + 42.1 • c = 6; -6 • c = 9.2; -9.2 • c = 8.4; -8.4

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

  5. Algebraic Proof of the Pythagorean Theorem Write two different expressions of the area of the large square and then set them equal to each other. This is just one of the many proofs of the Pythagorean Theorem!!

  6. Definition • Pythagorean triple– a set of positive integers, a, b, and c that satisfy the equation a2 + b2 = c2. • Example: 3, 4, 5 • 32 + 42 = 52.

  7. Example 1 • Find x. 5, 12, and 13 form a Pythagorean triple!!

  8. Example 2 • Find x.

  9. Try This! • Find the length of the missing leg.

  10. Example 3 • Find the area of the triangle. First we need to solve for h. Now we can find the area by using A = 1/2bh.

  11. Summarizer • Find the length of the second leg of a right triangle with a hypotenuse of 18 and a leg of 12.

  12. Homework

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