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Slide 1 11-3 The Pythagorean Theorem and the Distance Formula Special Right Triangles Converse of the Pythagorean Theorem The Distance Formula: An Application of the Pythagorean Theorem

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Slide 2 Parts of a Right Triangle Hypotenuse – side opposite of the right angle. Legs – are the other two sides.

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Slide 3 Pythagorean Theorem Given a right triangle with legs a and b and hypotenuse c, then c 2 = a 2 + b 2. If BC = 3 cm and AC = 4 cm, what is the length of AB? Answer: 4 = 3 = 5

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Slide 4 Proof of the Pythagorean Theorem

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Slide 5 Proof of the Pythagorean Theorem The square on one leg that is labeled 1 could be cut off and placed in the dashed space on the square of the hypotenuse. Then pieces 2, 3, 4, and 5 could be cut off and placed around piece 1 so that the square on the hypotenuse is filled exactly with five pieces. This shows that the sum of the areas of the squares on the two legs of a right triangle is equal to the area on the square of the hypotenuse.

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Slide 6 Use the Pythagorean theorem to find the height of the trapezoid: 5 2 + h 2 = 13 2 25 + h 2 = 169 h 2 = 169 – 25 h 2 = 144 h = = 12 Example Find the area of the trapezoid. Solution h A = ½ h (b 1 + b 2 ) A = ½(12)(10 + 18) A = 168 cm 2 The length of the bottom base is 5 + 10 + 3 = 18 cm. So the area of the trapezoid is 168 cm2

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Slide 7 Special Right Triangles 45 ° -45 ° -90 ° right triangle The length of the hypotenuse in a 45° -45° -90° right triangle is times the length of a leg.

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Slide 8 30 ° -60 ° -90 ° right triangle In a 30°-60°-90° right triangle, the length of the hypotenuse is twice the length of the leg opposite the 30° (the shorter leg. The leg opposite the 60° angle (the longer leg) is times the length of the shorter leg.

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Slide 9 Converse of the Pythagorean Theorem If ∆ABC is a triangle with sides of lengths a, b, and c such that c 2 = a 2 + b 2, then ∆ABC is a right triangle with the right angle opposite the side of length c. Example Determine if a triangle with the lengths of the sides given is a right triangle. a.20, 21, 29 b.5, 18, 25 Answer: Yes, 20 2 + 21 2 = 841 = 29 2 Answer: No, 5 2 + 18 2 = 349 25 2

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Slide 10 The Distance Formula: An Application of the Pythagorean Theorem

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Slide 11 The Distance Formula The distance between the points A(x 1, y 1 ) and B(x 2, y 2 ) is

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Slide 12 Example Determine what kind of triangle is formed by joining the points A(4, 7), B(–4, –7), and C(–7, 4). Solution So ∆ABC is an isosceles right triangle.

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Slide 13 HOMEWORK 11-3 Pages 781 – 783 # 1, 3, 7, 9, 17

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