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8.1 Pythagorean Theorem & Converse

8.1 Pythagorean Theorem & Converse. Essential Question: How do you use the Pythagorean Theorem and its converse?. The Pythagorean Theorem is named for Pythagoras, a Greek mathematician who lived in the sixth century B.C.

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8.1 Pythagorean Theorem & Converse

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  1. 8.1 Pythagorean Theorem & Converse Essential Question: How do you use the Pythagorean Theorem and its converse?

  2. The Pythagorean Theorem is named for Pythagoras, a Greek mathematician who lived in the sixth century B.C. • We now know that the Babylonians, Egyptians, and Chinese were aware of this relationship before its discovery by Pythagoras. • There are many proofs of this theorem. • Pythagoras created a school.

  3. Theorem 8-1 • 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

  4. A Pythagorean triple is a set of nonzero whole numbers a, b, and c that satisfy the equation a2 + b2 = c2. • These can help you solve problems faster, so be on the lookout for triples! • Here are some: • 3,4,5 • 5,12,13 • 8,15,17 • 7,24,25

  5. Example 1 and Practice 1

  6. Example 2 and Practice 2

  7. You can use the Converse of the Pythagorean Theorem to determine whether the triangle is right, obtuse, or acute. • Theorem 8-2 • 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. • If c2 = a2 + b2, then it is a right triangle.

  8. Example 4 and Practice 4

  9. Theorem 8-3 • 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, then the triangle is obtuse.

  10. Theorem 8-4 • If the square 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, then the triangle is acute.

  11. Example 5 and Practice 5

  12. Reflection • As we learned, Pythagoras was a philosopher as well as a mathematician. • Remember philosophy is the study of knowledge, reality, and existence. • It was common for mathematicians to also dive into the field of philosophy. • Why do you think this happened?

  13. Summary • Answer the essential question in detailed, complete sentences. • How do you use the Pythagorean Theorem and its converse? • Write 2-4 Study Questions in the left column to correspond to the notes.

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