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9.8 The Pythagorean Theorem and SPACE FIGURES

Advanced Geometry . 9.8 The Pythagorean Theorem and SPACE FIGURES. To find the length of diagonals in cubes and rectangular solids, a formula can be applied. The formula is really a DOUBLE USE of the Pythagorean Theorem!. √ a 2 + b 2. b. B. C. A. a. D. √ a 2 + b 2. c.

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9.8 The Pythagorean Theorem and SPACE FIGURES

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  1. Advanced Geometry 9.8 The Pythagorean Theorem and SPACE FIGURES

  2. To find the length of diagonals in cubes and rectangular solids, a formula can be applied. The formula is really a DOUBLE USE of the Pythagorean Theorem! √a2 + b2 b B C A a D √a2 + b2 c BASE: Triangle #1 d F G b (b = width) E a H (a = length)

  3. To find the length of diagonals in cubes and rectangular solids, a formula can be applied. The formula is really a DOUBLE USE of the Pythagorean Theorem! d √a2 + b2 B c C (c = height) A D √a2 + b2 c c INSIDE: Triangle #2 d 2 + c2 √a2 + b2 d = √a2 + b2 + c2 d2 = F G OR b d2 = a2 + b2 + c2 E a H SO

  4. To find the length of diagonals in cubes and rectangular solids, the following formula can be applied. Where: a = length, b = width, c = height d = √a2 + b2 + c2 d = √l2 + w2+ h2

  5. Find the length of the diagonal. 5 d (25 • 2) d = √l2 + w2+ h2 d = √(4)2 + (3)2 + (5)2 d = √16+ 9+ 25 d = √50 d = 5√2 3 4

  6. Find the length of the diagonal – check. Inside triangle diagonal 5 d Base triangle diagonal = √42 + 32 = √52 + 52 = √2 • 52 = 5√2 = √16 + 9 = √25 = 5 5 5 3 √ 4 This is the diagonal of the solid!

  7. Find the missing measure. 17 9 d = √l2 + w2+ h2 w 12 HINT! Substitute the measures you know, and then solve algebraically to find the missing measure!

  8. Find the missing measure. d = 17, l = 12, w = w, h = 9 17 = √122 + w2+ 92 289 = 144 + w2+ 81 2 2 17 9 9 289 = w2+ 225 √208 √16•13 4√13 289 - 225 = w2 w 8 17 = √122 + w2+ 92 w 8 64 = w2 12 CHECK: ? √ √ 122 + 82= 172 – 92 w = 8 ? 144 + 64 = 289 – 81 √ 208 = 208

  9. Find the missing measure. e 6√3 d = √l2 + w2+ h2 e e

  10. Find the missing measure of the cube. substitute 2 2 simplify e 6√3 square sides 6√3 = √3e2 6√3 = √3e2 d = √l2 + w2+ h2 d = √e2 + e2+ e2 e e 36•3 = 3e2 divide by 3 √ √ 36 = e2 take sq root e = 6

  11. 9.8 Assignment [The Pythagorean Theorem and Space Figures] P 414 (2; 5 – 9; 11, 12, 16)

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