1. Pythagoras Theorem c is the length of the hypotenuse (side opposite the right angle). a and b are the lengths of the other two sides. It does not matter whether a is the height and b the base or vice versa. b a c This theorem is based on right-angled triangles only. a 2 + b 2 = c 2

2. The next set of slides prove the theorem. Before the proof you need to know the following: 1)The area of a triangle = ½ base x height All the triangles below have the same area because the base and height are the same height base

3. height 1.We start with half the red square, which has Area = ½ base x height 2.We move one vertex while maintaining the base & height, so that the area remains the same. This is called a SHEAR. 3.We rotate this triangle, which does not change its area. base height 4.We mark the base and height for this triangle. (Area of green square)+ (Area of red square)= Area of the blue square The Pythagorean Theorem:

4. base height 5.We now do a shear on this triangle, keeping the same area. Remember that this pink triangle is half the red square. Half the red square. (Area of green square)+ (Area of red square)= Area of the blue square The Pythagorean Theorem: 1.We start with half the red square, which has Area = ½ base x height 2.We move one vertex while maintaining the base & height, so that the area remains the same. This is called a SHEAR. 3.We rotate this triangle, which does not change its area. 4.We mark the base and height for this triangle.

5. (Area of green square)+ (Area of red square)= Area of the blue square The Pythagorean Theorem: 6.The other half of the red square has the same area as this pink triangle, so if we copy and rotate it, we get this. So, together these two pink triangles have the same area as the red square. 7.We now take half of the green square and transform it the same way. Half the red square. We end up with this triangle, which is half of the green square. Half the green square. 9.Together, they have they same area as the green square. So, we have shown that the red & green squares together have the same area as the blue square. Shear Rotate Shear 8.The other half of the green square would give us this.

6. a 2 + b 2 = c 2 Replace letters 5 2 + 12 2 = c 2 Calculate 25 + 144 = c 2 169 = c 2 Swap sides c 2 = 169 c = √169 Use the √ button c = 13 Example 1 12 5 c

7. a 2 + b 2 = c 2 Replace letters 9 2 + b 2 = 15 2 Calculate 81 + b 2 = 225 Subtract 81 from both sides b 2 = 225 – 81 Calculate b 2 = 144 b = √144 Use the √ button b = 12 Example 2 b 9 15

8. a=4 b=8 Find side XY Look at red triangle WUY and find the hypotenuse a 2 + b 2 = c 2 Replace letters 4 2 + 8 2 = c 2 Calculate 16 + 64 = c 2 c 2 = 80 c = √80 = 8.94 c X Y U W 8.94

9. a=4 c Find side XY Look at the green triangle XWY and find the hypotenuse a 2 + b 2 = c 2 Replace letters 4 2 + 8.94 2 = c 2 Calculate 16 + 80 = c 2 c 2 = 96 c = √96 = 9.8 X Y U W b = 8.94

10. c=10 b=5 5 Find side WZ W Z X Y Look at green triangle WYX and find the height a 2 + b 2 = c 2 Replace letters a 2 + 5 2 = 10 2 Calculate a 2 + 25 = 100 a 2 = 100 – 25 a 2 = 75 a = √75 = 8.66 8.66 a

11. 55 Find side WZ W Z X Y Look at red triangle WYZ and find the hypotenuse a 2 + b 2 = c 2 Replace letters 8.66 2 + 10 2 = c 2 Calculate 75 + 100 = c 2 175 = c 2 c 2 = √ 175 c = √175 = 13.2 a=8.66 b=10

12. A 5 metre ladder is placed against a wall. The base is 1.5 metres from the wall. How high up the wall does the ladder reach? Draw a triangle and put in the numbers… Using a 2 + b 2 = c 2 a=1.5 c=5 1.5 2 + b 2 = 5 2 2.25 + b 2 = 25 b =  22.75 = 4.77m (to 2 d.p.) b b 2 = 25 – 2.25 = 22.75