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About 2,500 years ago, a Greek mathematician named Pythagorus discovered a special relationship between the sides of right triangles.

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Pythagorus realized that if you have a right triangle, 3 4 5

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and you square the lengths of the two sides that make up the right angle, 3 4 5

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and add them together, 3 4 5

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you get the same number you would get by squaring the other side. 3 4 5

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Is that correct? ? ?

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It is. And it is true for any right triangle. 8 6 10

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The two sides which come together in a right angle are called

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The lengths of the legs are usually called a and b. a b

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The side across from the right angle a b is called the

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a 2 + b 2 = c 2 Used to find a missing side of a right triangle a & b always shortest sides * c is always longest side

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Steps 1.Identify what sides you have and which side you are looking for. 2.Substitute the values you have into the appropriate places in the Pythagorean Theorem a 2 + b 2 = c 2 3.Do your squaring first… then solve the 2-Step equation. TOTD: if your answer under the radical is not a perfect square, leave your answer under the radical.

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4 5 c 6.40 c A. Pythagorean Theorem Substitute for a and b. a 2 + b 2 = c 2 4 2 + 5 2 = c 2 16 + 25 = c 2 41 = c Simplify powers. Solve for c; c = c 2. Example 1A: Find the Length of a Hypotenuse Find the length of the hypotenuse. 41 = c 2

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Example: 2 Finding the Length of a Leg in a Right Triangle 25 7 b 576 = 24 b = 24 a 2 + b 2 = c 2 7 2 + b 2 = 25 2 49 + b 2 = 625 –49 b 2 = 576 Solve for the unknown side in the right triangle. Pythagorean Theorem Substitute for a and c. Simplify powers.

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Try This: Example 1A 5 7 c A. Find the length of the hypotenuse. 8.60 c Pythagorean Theorem Substitute for a and b. a 2 + b 2 = c 2 5 2 + 7 2 = c 2 25 + 49 = c 2 74 = c Simplify powers. Solve for c; c = c 2.

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Try This: Example 2 b 11.31 12 4 b a 2 + b 2 = c 2 4 2 + b 2 = 12 2 16 + b 2 = 144 –16 b 2 = 128 128 11.31 Solve for the unknown side in the right triangle. Pythagorean Theorem Substitute for a and c. Simplify powers.

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15 = c B. Pythagorean Theorem Substitute for a and b. a 2 + b 2 = c 2 9 2 + 12 2 = c 2 81 + 141 = c 2 225 = c Simplify powers. Solve for c; c = c 2. Example 1B: Find the the Length of a Hypotenuse Find the length of the hypotenuse. triangle with coordinates (1, –2), (1, 7), and (13, –2)

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B. triangle with coordinates (–2, –2), (–2, 4), and (3, –2) x y The points form a right triangle. (–2, –2) (–2, 4) (3, –2) Try This: Example 1B Find the length of the hypotenuse. 7.81 c Pythagorean Theorem a 2 + b 2 = c 2 6 2 + 5 2 = c 2 36 + 25 = c 2 61 = c Simplify powers. Solve for c; c = c 2. Substitute for a and b.

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Example 3: Using the Pythagorean Theorem to Find Area a 66 44 a 2 + b 2 = c 2 a 2 + 4 2 = 6 2 a 2 + 16 = 36 a 2 = 20 a = 20 units ≈ 4.47 units Find the square root of both sides. Substitute for b and c. Pythagorean Theorem A = hb = (8)( 20) = 4 20 units 2 17.89 units 2 1212 1212 Use the Pythagorean Theorem to find the height of the triangle. Then use the height to find the area of the triangle.

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a 2 + b 2 = c 2 a 2 + 2 2 = 5 2 a 2 + 4 = 25 a 2 = 21 a = 21 units ≈ 4.58 units Find the square root of both sides. Substitute for b and c. Pythagorean Theorem A = hb = (4)( 21) = 2 21 units 2 4.58 units 2 1212 1212 Try This: Example 3 Use the Pythagorean Theorem to find the height of the triangle. Then use the height to find the area of the triangle. a 55 22

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Lesson Quiz 1. Find the height of the triangle. 2. Find the length of side c to the nearest meter. 3. Find the area of the largest triangle. 4. One leg of a right triangle is 48 units long, and the hypotenuse is 50 units long. How long is the other leg? 8m 12m 60m 2 14 units h c 10 m 6 m9 m Use the figure for Problems 1-3.

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