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Pythagorean Theorem 5.4

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Learn the Pythagorean Theorem. Define Pythagorean triple. Learn the Pythagorean Inequality. Solve problems with the Pythagorean Theorem and Inequality.

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The Pythagorean Theorem is probably the most famous mathematical relationship. It states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. a 2 + b 2 = c 2

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Find the value of x. Give your answer in simplest radical form. a 2 + b 2 = c 2 Pythagorean Theorem 2 2 + 6 2 = x 2 Substitute 2 for a, 6 for b, and x for c. 40 = x 2 Simplify. Find the positive square root. Simplify the radical.

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Find the value of x. Give your answer in simplest radical form. a 2 + b 2 = c 2 Pythagorean Theorem 4 2 + 8 2 = x 2 Substitute 4 for a, 8 for b, and x for c. 80 = x 2 Simplify. Find the positive square root. Simplify the radical.

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A set of three nonzero whole numbers a, b, and c such that a 2 + b 2 = c 2 is called a Pythagorean triple.

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Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. The side lengths are nonzero whole numbers that satisfy the equation a 2 + b 2 = c 2, so they form a Pythagorean triple. a 2 + b 2 = c 2 Pythagorean Theorem 14 2 + 48 2 = c 2 Substitute 14 for a and 48 for b. 2500 = c 2 Multiply and add. 50 = c Find the positive square root.

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Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. a 2 + b 2 = c 2 Pythagorean Theorem 8 2 + 10 2 = c 2 164 = c 2 The side lengths do not form a Pythagorean triple because is not a whole number.

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Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. The side lengths are nonzero whole numbers that satisfy the equation a 2 + b 2 = c 2, so they form a Pythagorean triple. a 2 + b 2 = c 2 Pythagorean Theorem 24 2 + b 2 = 26 2 Substitute 24 for a and 26 for c. b 2 = 100 Multiply and subtract. b = 10 Find the positive square root.

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Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. No. The side length 2.4 is not a whole number.

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The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths.

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You can also use side lengths to classify a triangle as acute or obtuse. A B C c b a

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To understand why the Pythagorean inequalities are true, consider ∆ABC.

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The third side of a triangle is less than the sum and greater than the difference of the other two sides. Remember!

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Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 5, 7, 10 Since 2< 10 < 12; 5, 7, and 10 can be the side lengths of a triangle. Classify the triangle. Since c 2 > a 2 + b 2, the triangle is obtuse. Add and compare. 100 > 74 Multiply. Compare c 2 to a 2 + b 2. Substitute the longest side for c. c 2 = a 2 + b 2 ? 10 2 = 5 2 + 7 2 ? 100 = 25 + 49 ?

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Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. Determine if the measures form a triangle. 5, 8, 17 Since 3 < 17 < 13, these cannot be the side lengths of a triangle.

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Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 7, 12, 16 Since 5 < 16 < 19; 7, 12, and 16 can be the side lengths of a triangle. Classify the triangle. Since c 2 > a 2 + b 2, the triangle is obtuse. Add and compare. 256 > 193 Multiply. Compare c 2 to a 2 + b 2. Substitute the longest side for c. c 2 = a 2 + b 2 ? 16 2 = 12 2 + 7 2 ? 256 = 144 + 49 ?

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Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. Determine if the measures form a triangle. 11, 18, 34 Since 7 < 34 < 29 ; these cannot be the sides of a triangle.

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Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 3.8, 4.1, 5.2 Since.3< 5.2 < 7.9; 3.8, 4.1, and 5.2 can be the side lengths of a triangle. Classify the triangle. Since c 2 < a 2 + b 2, the triangle is acute. Add and compare. 27.04 < 31.25 Multiply. Compare c 2 to a 2 + b 2. c 2 = a 2 + b 2 ? 5.2 2 = 3.8 2 + 4.1 2 ? 27.04 = 14.44 + 16.81 ? Substitute the longest side for c.

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Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 13; yes; the side lengths are nonzero whole numbers that satisfy Pythagorean’s Theorem.

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Tell if the measures 7, 11, and 15 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. yes; obtuse

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Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. a. b. Yes a triangle can be formed 1 < 6 < 9 Acute Yes a triangle can be formed 16 < 26 < 36 Right c. Yes a triangle can be formed 5 < 25 < 35 Right

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Find the exact answer of the missing side. a. b. c. d. e. f.

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a. b. c. x = 25 x = 34 d. e. f. Find the exact answer of the missing side.

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a. b. c. x = 15 x = 51 d. e. f. x = 60 x = 30 Find the exact answer of the missing side.

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a. b. c. x = 15 x = 51 d. e. f. Find the exact answer of the missing side. x = 82

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Assignment Section 5.4 8 - 30

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