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 90o angle 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 90o right angle Level 7 Level 8

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!!

Using Pythagoras’ Theorem Area C c2 So what is Pythagoras’ Theorem? He said that: Area A a2 a b c a2 + b2 = c2 Area B b2 “For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.” Pythagoras

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

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

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

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

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

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

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

Pythagoras’ Theorem Answer = 75miles Answer = 27.7m 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

Pythagoras’ Theorem Answer = 3.7m Answer = 9.5m 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