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8-1 The Pythagorean Theorem and Its Converse

8-1 The Pythagorean Theorem and Its Converse. Parts of a Right Triangle. In a right triangle, the side opposite the right angle is called the hypotenuse . It is the longest side. The other two sides are called the legs. The Pythagorean Theorem.

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8-1 The Pythagorean Theorem and Its Converse

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  1. 8-1 The Pythagorean Theorem and Its Converse

  2. Parts of a Right Triangle • In a right triangle, the side opposite the right angle is called the hypotenuse. • It is the longest side. • The other two sides are called the legs.

  3. The Pythagorean Theorem Pythagorean Theorem: If a triangle is a right triangle, then 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. Pythagorean Triples • A Pythagorean triple is a set of nonzero whole numbers that satisfy the Pythagorean Theorem. • Some common Pythagorean triples include: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 • If you multiply each number in the triple by the same whole number, the result is another Pythagorean triple!

  5. Finding the Length of the Hypotenuse • What is the length of the hypotenuse of ABC? Do the sides form a Pythagorean triple?

  6.  The legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse? Do the sides form a Pythagorean triple?

  7. Finding the Length of a Leg • What is the value of x? Express your answer in simplest radical form.

  8.  The hypotenuse of a right triangle has length 12. One leg has length 6. What is the length of the other leg? Express your answer in simplest radical form.

  9. Triangle Classifications Converse of the Pythagorean Theorem: If the square of the length of the longest 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, than ABC is a right triangle. Theorem 8-3: If the square of the length of the longest side of a triangle is great than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse. • If c2 > a2 + b2, than ABC is obtuse. 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, then the triangle is acute. • If c2 < a2 + b2, than ABC is acute.

  10. Classifying a Triangle Classify the following triangles as acute, obtuse, or right. • 85, 84, 13 • 6, 11, 14 • 7, 8, 9

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