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Pythagorus realized that if you have a right triangle,

About 2,500 years ago, a Greek mathematician named Pythagorus discovered a special relationship between the sides of right triangles. 5. 3. 4. Pythagorus realized that if you have a right triangle,. 5. 3. 4. and you square the lengths of the two sides that make up the right angle,. 5. 3.

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Pythagorus realized that if you have a right triangle,

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

  2. 5 3 4 Pythagorus realized that if you have a right triangle,

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

  4. 5 3 4 and add them together,

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

  6. Is that correct? ? ?

  7. 10 8 6 It is. And it is true for any right triangle.

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

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

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

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

  12. The side across from the right angle is called the hypotenuse. a b

  13. a2 + b2 = c2 Pythagorean Theorem #10 Used to find a missing side of a right triangle a & b always shortest sides * c is always longest side

  14. Steps • Identify what sides you have and which side you are looking for. • Substitute the values you have into the appropriate places in the Pythagorean Theorem a2 + b2 = c2 • 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.

  15. 41 = c Solve for c; c = c2. Example 1A: Find the Length of a Hypotenuse Find the length of the hypotenuse. c A. 4 5 Pythagorean Theorem a2 + b2 = c2 42 + 52 = c2 Substitute for a and b. Simplify powers. 16 + 25 = c2 41 = c2 6.40c

  16. 576 = 24 Example: 2 Finding the Length of a Leg in a Right Triangle Solve for the unknown side in the right triangle. Pythagorean Theorem a2 + b2 = c2 25 Substitute for a and c. 72 + b2 = 252 b Simplify powers. 49 + b2 = 625 –49 –49 b2 = 576 7 b = 24

  17. 74 = c Solve for c; c = c2. Try This: Example 1A Find the length of the hypotenuse. c A. 5 7 Pythagorean Theorem a2 + b2 = c2 52 + 72 = c2 Substitute for a and b. Simplify powers. 25 + 49 = c2 8.60c

  18. 128  11.31 Try This: Example 2 Solve for the unknown side in the right triangle. a2 + b2 = c2 Pythagorean Theorem 12 Substitute for a and c. b 42 + b2 = 122 Simplify powers. 16 + b2 = 144 –16 –16 4 b2 = 128 b 11.31

  19. Solve for c; c = c2. 225 = c Example 1B: Find the the Length of a Hypotenuse Find the length of the hypotenuse. triangle with coordinates B. (1, –2), (1, 7), and (13, –2) Pythagorean Theorem a2 + b2 = c2 Substitute for a and b. 92 + 122 = c2 Simplify powers. 81 + 141 = c2 15= c

  20. y x Solve for c; c = c2. 61 = c Try This: Example 1B Find the length of the hypotenuse. B. triangle with coordinates (–2, –2), (–2, 4), and (3, –2) (–2, 4) The points form a right triangle. a2 + b2 = c2 Pythagorean Theorem 62 + 52 = c2 Substitute for a and b. 36 + 25 = c2 Simplify powers. (3, –2) (–2, –2) 7.81c

  21. a = 20 units ≈ 4.47 units 1 2 1 2 A = hb = (8)( 20) = 4 20 units2 17.89 units2 Example 3: Using the Pythagorean Theorem to Find Area Use the Pythagorean Theorem to find the height of the triangle. Then use the height to find the area of the triangle. a2 + b2 = c2 Pythagorean Theorem Substitute for b and c. a2 + 42 = 62 a2 + 16 = 36 6 6 a a2 = 20 4 4 Find the square root of both sides.

  22. a = 21 units ≈ 4.58 units 1 2 1 2 A = hb = (4)( 21) = 2 21 units2 4.58 units2 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. a2 + b2 = c2 Pythagorean Theorem a2 + 22 = 52 Substitute for b and c. 5 5 a2 + 4 = 25 a a2 = 21 2 2 Find the square root of both sides.

  23. Lesson Quiz Use the figure for Problems 1-3. 1. Find the height of the triangle. 8m 2. Find the length of side c to the nearest meter. c 10 m h 12m 3. Find the area of the largest triangle. 6 m 9 m 60m2 4. One leg of a right triangle is 48 units long, and the hypotenuse is 50 units long. How long is the other leg? 14 units

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