Using Pythagoras’ Theorem

1. Using Pythagoras’ Theorem L.O. Use Pythagoras Theorem to find missing sides on a triangle Solve real-life problems using Pythagoras Theorem We are learning this because… • 3000 years ago the Egyptians used Pythagoras Theorem to build the Great Pyramids using knotted rope to make a 90oangle using a 3,4,5 triangle. Today builders using pieces of wood with length 3ft, 4ft, 5ft to the same thing to get a perfect 90oright angle Level 7 Level 8

2. Pythagoras’ Theorem I was born at Samos, in Greece, and lived from 580 to 500 B.C. I was a Mathematician who became famous for discovering something unique about right – angled triangles. Now you are going to try to find out what I discovered!!

3. Using Pythagoras’ Theorem c a Area C c2 a2 + b2 = c2 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 b Area A a2 Area B b2

4. Using Pythagoras’ Theorem Area C 9 +16 = 25 Find the Length of side x We can use Pythagoras’ Theorem to find the longest side in a right –angled triangle We SQUARE to get the area of the smaller squares We ADD to get the area of the biggest square How do we get the length of side x We SQUARE ROOT the area to get the length of side x x =25 = 5cm Area A 32 = 9 x 3cm 4cm Area B 42 = 16

5. Using Pythagoras’ Theorem Level 7 Example 1 Find the Length of side x We can use Pythagoras’ Theorem to find the longest side in a right –angled triangle 2. Add Square x= 130 • Square 92 = 81 x 7cm 72 = 49 9cm x2 = 130 Root x = 11.4cm

6. Using Pythagoras’ Theorem Level 7 Example 2 Find the Length of side x We can use Pythagoras’ Theorem to find the longest side in a right –angled triangle 2. Add Square x= 80 • Square 82 = 64 x 4cm 42 = 16 8cm x2 = 80 Root x = 8.9

7. Using Pythagoras’ Theorem Level 7 Example 3 Find the Length of side x We can use Pythagoras’ Theorem to find a Short side in a right –angled triangle 2. Subtract Square x= 95 • Square 122 = 144 12cm x 72 = 49 7cm x2= 95 Root x = 9.7cm

8. Using Pythagoras’ Theorem Level 7 Example 4 Find the Length of side x We can use Pythagoras’ Theorem to find a Short side in a right –angled triangle 2. Subtract Square x= 304 • Square 232 = 529 23mm 15mm 152 = 225 x x2= 304 Root x = 17.4cm

9. Using Pythagoras’ Theorem 19m 60cm For each of the following triangles, calculate the length of the missing side, giving your answers to one decimal place when needed. 1.1cm 6cm 14m 1.5cm 3 2 1 25cm 3cm 13mm 11cm Answer = 23.6m Answer = 9.8cm Answer = 6.7cm 6 5 4 5cm 12mm Answer = 65cm Answer = 1.0cm Answer = 5mm Level 7

10. Using Pythagoras’ Theorem 8 If a right angle has short lengths 14cm and 8cm, what is the length of the longest side. Calculate the length of the diagonal of this square. 7 6cm Answer = 16.1cm Answer = 8.5cm 10 9 Calculate the height of this isosceles triangle. Calculate the base of this isosceles triangle. Answer = 12cm 10cm 10cm Answer = 11.3cm 12cm 12cm 8cm 8cm

11. Pythagoras’ Theorem Level 8 Real Life Problem 1 • A boat travels 45 miles east then 60 miles north, how far is it from where it started? (hint: draw a diagram) Answer = 75miles Real Life Problem 2 A swimming pool is 25m by 12m, if someone swam from one corner to the other, how far would they have swam?(hint: draw a diagram) Answer = 27.7m

12. Pythagoras’ Theorem Level 8 Real Life Problem 3 • A ladder which is 4m long leans against a wall, the bottom of the ladder is 1.5m from the bottom of the wall, how high up the wall does the ladder go? (hint: draw a diagram) Answer = 3.7m Real Life Problem 4 A rope of length 10m is stretched from the top of a pole 3m high until it reaches the ground. How far is the end of the rope to the base of the pole.(hint: draw a diagram) Answer = 9.5m