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ESSENTIAL OBJECTIVE Use the Pythagorean Theorem and the Distance Formula.

ESSENTIAL OBJECTIVE Use the Pythagorean Theorem and the Distance Formula. Right Triangle. The sides that form the right angle are called the legs . The side opposite the right angle is called the hypotenuse. The Pythagorean Theorem.

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ESSENTIAL OBJECTIVE Use the Pythagorean Theorem and the Distance Formula.

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  1. ESSENTIAL OBJECTIVE Use the Pythagorean Theorem and the Distance Formula.

  2. Right Triangle • The sides that form the right angle are called the legs. • The side opposite the right angle is called the hypotenuse.

  3. The Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

  4. The Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

  5. Example 1 SOLUTION (hypotenuse)2 = (leg)2 + (leg)2 c2 = 169 Find the Length of the Hypotenuse Find the length of the hypotenuse. c2 = 52 + 122 c2 = 25 + 144 c2 = 169 c = 13 The length of the hypotenuse is 13. ANSWER

  6. Example 2 SOLUTION (hypotenuse)2 = (leg)2 + (leg)2 b2 147 = Find the Length of a Leg Find the unknown side length. 142 = 72 + b2 196 = 49 + b2 196 – 49 = 49 + b2 – 49 147 = b2 12.1≈ b The side length is about 12.1. ANSWER

  7. Find the unknown side length. I DO…..! 1.

  8. Checkpoint Find the Lengths of the Hypotenuse and Legs Find the unknown side length. I DO…..! 1.

  9. WE DO….! 2.

  10. Checkpoint Find the Lengths of the Hypotenuse and Legs WE DO….! 2.

  11. 3. YOU DO….!

  12. Checkpoint 3. Find the Lengths of the Hypotenuse and Legs YOU DO….!

  13. Example 3 SOLUTION Using the Pythagorean Theorem. (AB)2 = 25 the positive square root. Find the Length of a Segment Find the distance between the points A(1,2) and B(4,6). (hypotenuse)2 = (leg)2 + (leg)2 (AB)2 = 32 + 42 (AB)2 = 9 + 16 (AB)2 = 25 AB = 5

  14. The Distance Formula • If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the distance between A and B is AB = (x2 - x1)2 + (y2 - y1)2

  15. Example 4 (x2 – x1)2 + (y2–y1)2 DE = The Distance Formula (3 – 1)2 + (–2–2)2 = Substitute. 22 + (–4)2 = Simplify. 4 + 16 = Multiply. 20 = Add. Use the Distance Formula Find the distance between D(1, 2) and E(3, 2). SOLUTION Begin by plotting the points in a coordinate plane. x1 = 1,y1 = 2,x2 = 3,andy2 = –2. Approximate with a calculator. The distance between D and E is about 4.5 units. ≈ 4.5 ANSWER

  16. Example 4 Use the Distance Formula Find the distance between D(1, 2) and E(3, 2). The distance between D and E is about 4.5 units. ANSWER

  17. Checkpoint Find the distance between the points. Use the Distance Formula 4.

  18. Checkpoint Use the Distance Formula 5.

  19. Find the value of x. Tell what theorem(s) you used. 1. 2. x =90; Base Angles Theorem, Triangle Sum Theorem ANSWER REVIEW x = 70;Base Angles Theorem ANSWER

  20. 3. 4. Find the value of x. x = 4 ANSWER x =9 ANSWER

  21. HW Practice 4.4A

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