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Pythagoras’ Theorem

Pythagoras’ Theorem. If x 2 = 100, what is x?. 100. x 2. =. x. =. = 10. 100. If x 2 = 50, what is x?. 50. x 2. =. x. =. = 7.07 (2 d.p. ). 50. Area C. Area A. Area B. 9. 16. 25. C. A. 3. 4. B. Area C. Area A. Area B. 9. 16. 25. 16. 64. 80. C. A. 8. 4. B.

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Pythagoras’ Theorem

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  1. Pythagoras’ Theorem

  2. If x2 = 100, what is x? 100 x2 = x = = 10 100 If x2 = 50, what is x? 50 x2 = x = = 7.07 (2 d.p.) 50

  3. Area C Area A Area B 9 16 25 C A 3 4 B

  4. Area C Area A Area B 9 16 25 16 64 80 C A 8 4 B

  5. 3 5 Area C Area A Area B 6 9 16 25 5 16 64 80 25 25 50 9 36 45 36 64 100 7 49 64 113 8 8 6

  6. How do you work out the area of the squares you draw at the a and the b sides of the triangle? c x c Call the 3rd side (the longest) c. c2 b x b c What will the area of the square drawn at this side be? b b2 a Can you now find a rule that connects the areas of the 3 squares and the 3 sides of the triangle? a x a a2 a2 b2 = c2 +

  7. a2 b2 = c2 + HYPOTENUSE PYTHAGORAS' THEOREM c b “The square of the hypotenuse is equal to the sum of the squares of the other two sides.” a

  8. Pythagoras of Samos 580BC-500BC

  9. a2 + b2 = c2 122 + 162 = x2 x2 144 + 256 = x 12cm x2 400 = = x 400 16cm 20cm = x

  10. a2 + b2 = c2 92 + 72 = x2 x2 81 + 49 = x 9cm x2 130 = = x 130 7cm 11.4cm = x

  11. a2 + b2 = c2 8.22 + x2 = 12.32 x 8.2cm 151.29 67.24 + x2 = 84.05 x2 = 12.3cm x = 84.05 x = 9.17cm

  12. a2 + b2 = c2 152 + x2 = 182 18m 324 225 + x2 = x 99 x2 = x = 99 15m x = 9.95m

  13. a2 b2 = c2 + HYPOTENUSE c PYTHAGORAS' THEOREM b a

  14. Finding the hypotenuse? • Square • Add • Square root Finding a shorter side? • Square • Subtract • Square root

  15. a2 + b2 = c2 82 + 92 = x2 x2 64 + 81 = x2 145 = = x 145 12.04cm = x a2 + b2 = c2 x2 + 142 = 172 289 x2 + 196 = x2 = 93 x = 93 x = 9.64mm

  16. b a c a2 + b2 = c2 4.82 + 7.52 = c2 c2 56.25 23.04 + = c2 79.29 = = c 79.29 8.9km = c

  17. Find the area of this equilateral triangle Area of a triangle = ½ base x height h a2 + b2 = c2 h2 + 92 = 182 18 324 h2 + 81 = h h2 = 243 h = 243 9 h = 15.59cm Area = ½ x 18 x 15.59 Area = 140.3cm2

  18. Finding the hypotenuse? • Square • Add • Square root Finding a shorter side? • Square • Subtract • Square root

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