1. PYTHAGORAS Aim: To be able to know Pythagoras’ Theorem All: Will be able to recall theorem. Most: Will be able to use to find the length of hypotenuse. Some: Will be able to use it to find the length of the shorter side.

2. Study Plus Pythagoras’ Theorem Trigonometry Polygon angles Ratio & Proportion

3. Pythagoras’ Theorem Only works in right angled triangles Nothing to do with angles Two main types of questions Type 1 Type 2

4. Hypotenuse

5. The hypotenuse is the longest side in a right angled triangle. It is always the side opposite the right angle. h y p o t e n u s e

6. Spotting the Hypotenuse hypotenuse

7. Pythagoras’ Theorem ‘In a right-angled triangle, the area of the square on the side opposite to the right angle is equal to the sum of the squares on the sides forming the right angle.’

8. Pythagoras’ Theorem Pythagoras’ Theorem states that: ‘The sum of the squares of the lengths of the sides containing the right angle is equal to the square of the hypotenuse.’ In other words : a 2 + b 2 = c 2 a b c A B C ‘c’ must be the hypotenuse You must square the numbers first, and then add Remember that ‘square’ means to multiply the number by itself (3 2 = 3x3 = 9)

9. Type 1 (Finding The Hypotenuse) a² + b ² Square, square Add Square root 10 18 ? Find the missing side. 10 2 = 100, 18 2 = 324 100 + 324 = 424 424 = 20.6 (3 sf)

10. Type 1 Find the missing sides. Give your answers to 3 sf. 8cm 10cm 11m 7m 24km 5km 10 2 = 100, 8 2 = 64 100 + 64 = 164 164 = 12.8cm 11 2 = 121, 7 2 = 49 121 + 49 = 170 170 = 13.0m 24 2 = 576, 5 2 = 25 576 + 25 = 601 601 = 24.5km

11. Type 2 (Finding A Leg) Square, square Take away Square root Find the missing side. 3.1 2 = 9.61, 2 2 = 4 9.61 – 4 = 5.61 5.61 = 2.37 miles (3 sf) 2 miles 3.1 miles ?

12. Type 2 Find the missing sides. Give your answers to 3 sf. 9cm 15cm 13m 6m 24km 9km 15 2 = 225, 9 2 = 81 225 – 81 = 144 144 = 12.0cm 13 2 = 169, 6 2 = 36 169 – 36 = 133 133 = 11.5m 24 2 = 576, 9 2 = 81 576 – 81 = 495 495 = 22.2km

13. Navigation (1) Navigation problems are often solved using Pythagoras’ Theorem. Make sure you know which way North, South, East and West point! N S EW

14. Navigation (2) A plane leave an airport and travels 32km west then it turns and travels 41km north. It develops a problem and has to return to the airport. How far is it? Step 1. Draw a diagram 32km Airport ? Step 2. Use Pythagoras This is Type 1. We have to find the hypotenuse. 41km 32 2 = 1024, 41 2 = 1681 1024 + 1681 = 2705 2705 = 52.0km

15. Word Problems (1) Sometimes it is not obvious that you need to use Pythagoras’ Theorem. If you draw a diagram you might spot a right angled triangle you can use…

16. Word Problems (2) Farmer Giles wants to cross to the diagonally opposite corner of his rectangular marrow field. The field measures 400m by 500m. How much distance does he save by going across the field rather than going around it? Step 1. Draw a diagram 500m 400m ?

17. Word Problems (3) Step 2. Use Pythagoras 400 500 500 2 = 250 000, 400 2 = 160 000 250 000 + 160 000 = 410 000 410000 = 640.3m 640.3 Step 3. The final answer Distance round outside = 500 + 400 = 900m. So Farmer Giles saves 900 – 640.3 = 260m (3 sf)

18. Word Problems (4) A ladder rests against a wall. For safety reasons the base of the ladder must be at least 2m from the wall. The ladder is 6.2m long. How high up the wall can the ladder reach? Step 1. Draw a diagram ? 2m 6.2m

19. Word Problems (5) Step 2. Use Pythagoras 2m 5.87m This is a Type 2 problem. 6.2 2 = 38.44, 2 2 = 4 34.44 = 5.87m (3 sf) 38.44 – 4 = 34.44 6.2m

20. And finally …. Pythagoras’ Theorem only works for a particular type of triangle, which type? If you are finding the hypotenuse, do you add or subtract the shorter sides squared? What is meant by “hypotenuse”? I wish I had worked harder at school!