1. PYTHAGORAS THEOREM Pythagoras Theorem

3. a b c “C “ is the longest side of the triangle “a” and “b” are the two other sides a 2 +b 2 =c 2

4. Draw a DEF with l(DE) =3 cm, m E = 90 0, l(EF)= 4 cm Measure l(DF) it is 5cm. 3cm 4cm 5cm D E F

5. 4cm 5cm D E F Q S 3 cm Square DESQ of side 3 cm drawn on side DE. So the Area of DESQ = Side * Side = 3 * 3 = 9 Sq.cm

6. 5cm D E F Q S X Y 4 cm 3 cm Square EFYX of side 4cm drawn on side EF. So the Area of EFYX = Side * Side, = 4 * 4 = 16 Sq.cm

7. 5cm D E F Q S X Y 4 cm 3 cm Square DFMN = 5 * 5 = 25 sq.cm M N

8. D E F Q S XY 4 CM 3 CM We add the area of square DESQ & EFYX We find that their sum is also 25 sq.cm. That is 3 2 + 4 2 = 5 2 E

9. Now you can use algebra to find any missing value, as in the following examples:algebra Example: Solve this triangle. a 2 + b 2 = c 2 5 2 + 12 2 = c 2 25 + 144 = c 2 169 = c 2 c 2 = 169 c = √169 c = 13

11. Example: What is the diagonal distance across a square of size 1?

12. Pythagorean Theorem Algebra Proof

13. The Pythagorean Theorem states that, in a right triangle, the square of a (a 2 ) plus the square of b (b 2 ) is equal to the square of c (c 2 ): a 2 + b 2 = c 2

14. We can show that a 2 + b 2 = c 2 using Algebra The diagram has that "abc" triangle in it (four of them actually):

15. Area of Whole Square It is a big square, with each side having a length of a+b, so the total area is: A = (a+b)(a+b) Area of Whole Square: It is a big square, with each side having a length of a+b, so the total area is: A = (a+b)(a+b) Area of The Pieces : Now let's add up the areas of all the smaller pieces:

16. First, the smaller (tilted) square has an area of A = c 2 And there are four triangles, each one has an area of A =½ab So all four of them combined is A = 4(½ab) = 2ab So, adding up the tilted square and the 4 triangles gives: A = c 2 +2ab

17. Both Areas Must Be Equal: The area of the large square is equal to the area of the tilted square and the 4 triangles. This can be written as: (a+b)(a+b) = c 2 +2ab NOW, l rearrange this to see if we can get the pythagoras theorem: Start with: (a+b)(a+b)=c 2 + 2ab Expand (a+b)(a+b): a 2 + 2ab + b 2 =c 2 + 2ab Subtract "2ab" from both sides: a 2 + b 2 =c2c2

18. Therefore “In a right angled triangle, the area of the square on the Hypotenuse is equal to the sum of area of the square on the two remaining sides “

19. It works the other way around, too: when the three sides of a triangle make a 2 + b 2 = c 2, then the triangle is right angled. Example: Does this triangle have a Right Angle? Does a 2 + b 2 = c 2 ? a 2 + b 2 = 10 2 + 24 2 = 100 + 576 = 676 c 2 = 26 2 = 676 They are equal, so... Yes, it does have a Right Angle!

20. Example: Does an 8, 15, 16 triangle have a Right Angle? Does 8 2 + 15 2 = 16 2 ? 8 2 + 15 2 = 64 + 225 = 289, but 16 2 = 256 So, NO, it does not have a Right Angle.

21. Therefore “In a right angled triangle, the area of the square on the Hypotenuse is equal to the sum of area of the square on the two remaining sides “