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## The Pythagorean Theorem

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**The Pythagorean Theorem**Geometry Mrs. Pam Miller January 2010**The Pythagorean Theorem**• Greek Mathematician, Pythagoras, proved this theorem. • Applies to right triangles. • Many different proofs exist, including one by President Garfield.**The Theorem**For any right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. hypotenuse leg leg2 + leg2 = hyp2 leg**The Abbreviated Version**We often see the Pythagorean Theorem stated as: a2 + b2 = c2 c a b**Some Examples**Find the value of x. ( Remember that the length of a segment must be a positive number.) leg2 + leg2 = hyp2 a) 32 + 72 = x2 x 3 9 + 49 = x2 7 58 = x2 √58 = √x2 √58 = x**Examples (cont’d.)**8 leg2 + leg2 = hyp2 b) x 82 + x2 = 102 10 64 + x2 = 100 64 - 64 + x2 = 100 - 64 X2 = 36 √x2 = √36 X = 6**Practice with Radicals**Work with the Pythagorean Theorem often requires us to work with radicals. Simplify each expression: A) (√3) 2 B) ( 3 √11 ) 2 3 √11 × 3 √ 11 √3 × √3 9 × √ 121 √9 9 × 11 3 99**Your Turn**Simplify each expression: E. A. (√5) 2 B. (2 √7) 2 F. C. (7 √2 ) 2 G. D. (2n) 2**The Answers**Simplify each expression: E. A. (√5) 2 = 5 = 9/5 B. (2 √7) 2 = 28 = 1/2 F. C. (7 √2 ) 2 = 98 G. = 24/9 D. (2n) 2 = 4n2**Pythagorean Triples**3, 4, 5 5, 12, 13 10, 24, 26 8, 15, 17 7, 24, 25 6, 8, 10 9, 12, 15 In every triple, the largest # is the length of the hypotenuse and the 2 smaller numbers are the lengths of the legs of the right triangle. 12, 16, 20 15, 20, 25**The Converse**If the square of 1 side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. More simply… if c2 = a2 + b2, then the triangle is a right triangle. c a b**Pythagorean Inequalities**If c2 > a2 + b2, then the triangle is obtuse. If c2 < a2 + b2, then the triangle is acute. c a b**Practice**The sides for 3 triangles are given. Decide if each triangle is acute, right or obtuse. A) 14, 7, 9 B) 2.5, 6, 6.5 C) 2, 3, 3.5 6.52 ___ 2.52 + 62 142 ___ 72 + 92 3.52 ___ 22 + 32 196 ___ 49 + 81 42.25 ___ 6.25 + 36 12.25 ___ 4 + 9 196 > 130 42.25 = 42.25 12.25 < 13 It’s obtuse! It’s right! It’s acute!**Check It Out**A triangle has sides with the following lengths: 9, 40, & 41. Is this a right triangle? Does 412 = 92 + 402 ? Does 1681 = 81 + 1600? YES !! So, the triangle is a right triangle.**More Practice**A right triangle has one leg with a length of 48 and a hypotenuse with a length of 80. What is the length of the other leg? 64 Here’s how: 482 + x2 = 802 x2 = 4096 2304 + x2 = 6400 √x2 = √ 4096 x2 = 6400 - 2304 x = 64**Practice (cont’d)**A triangle has side lengths of 7, 10, & 12. Is the triangle a right triangle? Use the Converse of the Pythagorean Theorem! Here’s How: Does c2 = a2 + b2 ? Does 122 = 72 + 102 ? Does 144 = 49 + 100? NO !**Practice (cont’d)**A triangle has side lengths of 8, 15, and 18. Is the triangle right, acute, or obtuse? Remember: If c2 < a2 + b2, you have an acute triangle. If c2 = a2 + b2, you have a right triangle. If c2 > a2 + b2, you have an obtuse triangle. Here’s How: 182 ____ 82 + 152 324 > 289 324 ____ 64 + 225 It’s obtuse!**Other Applications**Find the area of the figure. Leave your answer in radical form. 8 8 Area of a triangle = 1/2 bh 8 Use the Pythagorean Theorem to find “h”: h 8 42 + h2 = 82 h2 = 48 16 + h2 = 64 √ h2 = √ 48 4 h2 = 64 - 16 h = 4 √3 More….**Find the Area of the Triangle**Area = 1/2 bh 4 √3 8 Area = 1/2 (8) (4 √3) 8 8 Area = 16 √3 8 4**Last Problem!**Find the area of the square. Leave your answer in radical form. Use the Pythagorean Theorem to find the length of the sides (s). 6 s s2 + s2 = 62 √ s2 = √ 18 s s = 3 √2 2s2 = 36 So the area of the square is : 3 √2 × 3 √ 2 s2 = 18 = 9 √4 = 9 × 2 = 18**What Have You Learned?**• Pythagorean Theorem • Converse of the Pythagorean Theorem • Pythagorean Triples • Pythagorean Inequalities • Applications