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8-2 The Pythagorean Theorem and Its Converse
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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.
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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
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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!
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Finding the Length of the Hypotenuse
What is the length of the hypotenuse of ABC? Do the sides form a Pythagorean triple?
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The legs of a right triangle have lengths 10 and 24
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?
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Finding the Length of a Leg
What is the value of x? Express your answer in simplest radical form.
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The hypotenuse of a right triangle has length 12
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.
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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: 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: 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.
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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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