2. z Hypotenuse x y x 2 = y 2 + z 2 Right Angle Triangle x y x 2 > y 2 + z 2 Obtuse Triangle z xy x 2 < y 2 + z 2 z Acute Triangle Pythagoras

3. So what is Pythagoras’ Theorem? He said that: “For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.” Pythagoras a 2 + b 2 = c 2 a b c Area A a 2 Area B b 2 Area C c 2

4. Perigal’s Dissection The Theorem of Pythagoras: A Visual Demonstration In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. Henry Perigal (1801 – 1898) Gravestone Inscription

5. In fact any letters can be used in Pythagoras Theorem, the important thing is that the hypotenuse squared is equal to the other two sides squared. Hypotenuse a a 2 = b 2 + c 2 b 2 = a 2 + c 2 r 2 = s 2 + t 2 b c a b c r s t x y z x 2 = y 2 + z 2

6. We can use Pythagoras theorem to find missing sides in a right angle triangle. 10cm 6cm x x² = 6² + 10² x² = 36 + 100 x² = 136

7. 9cm 5cm x x² = 9² + 5² x² = 81 + 25 x² = 106

8. x x B A CD Find x Find the shortest distance of A to D 1. 2. 3. 8cm 11cm 12m 7m 12m 5m

9. So far we have used Pythagoras theorem to find the hypotenuse. 10cm 6cm x 10² = 6² + x² 100 = 36 + x² 64 = x² x = 8

10. 15cm 6cm x 15² = 6² + x² 225 = 36 + x² 194 = x² x =

11. x x B A C D Find x 1. 2. 3. Find AB 5cm 13cm 12m 7m 12m 5m 3m

12. Complete Questions 10-12, 13-15, 19-21, 36-38 Page 434

13. 6 7 x² = 7² + 6² x² = 49 + 36 x² = 85 x = 9.2cm A B

14. A B