The

1. The Pythagoras Theorem c a a2 + b2 = c2 b

2. This is a right triangle:

3. We call it a right triangle because it contains a right angle.

4. The measure of a right angle is 90o 90o

5. The little square in a triangle tells you it is a right angle. 90o

6. About 2,500 years ago, a Greek mathematician named Pythagoras discovered a special relationship between the sides of right triangles.

7. 5 3 4 Pythagoras realized that if you have a right triangle,

8. 5 3 4 and you square the lengths of the two sides that make up the right angle,

9. 5 3 4 and add them together,

10. 5 3 4 you get the same number you would get by squaring the other side.

11. 5 3 4 9 + 16 = 25 25 = 25 the sum of the squares of the lengths of the two sides. The square of the length of the hypotenuse of a right triangle

12. c a b Pythagoras Theorem states that The square of the length of the hypotenuse of a right triangle is the sum of the squares of the lengths of the two sides. Hypotenuse Hypotenuse is located opposite to the right angle in the triangle. Hypotenuse is also the longest side of the triangle.

13. PYTHAGOREAN TRIPLET: The 3 numbers which are satisfying the Pythagoras theorem are called the Pythagorean Triplets and can form right triangle using the given sides. We can also say given 3 numbers are pythagorean triplets if sum of square of two smallest numbers is equal to square of third number.

14. The numbers (6, 8, 10 ) are called a pythagorean triplets since they obey the rule a² + b² = c² Let’s check how, In the numbers 6 , 8 , 10 a = 6 , b = 8 and c = 10 Now substitute these values in our formula Remember: Always the longest side will be the hypotenuse. and It will be “c” in our formula a² + b² = c² 6² + 8² 10² 36 + 64 100 100 100 Since, the numbers (6, 8, 10 ) obey the rule a²+b² = c² they are “ Pythagorean Triplets “ =

15. Example 1 : Are 11, 12 and 5 a Pythagorean Triplet? Solution: Pythagorean Triplet : Sum of square of two smaller numbers = square of third (larger) number a² + b² = c² 52 + 112122 25 + 121 144 146 144 Ans:- Since sum of square of two small number is not equal to square of third number therefore the numbers 11,12 and 5 does not form a Pythagorean Triplet. In 11, 12 and 5, the smaller numbers are 5 and 11. a= 5 ,b = 11 and c=12 ≠

16. Example 2 : Are 5, 13 and 12 a Pythagorean Triplet? Solution: Pythagorean Triplet : Sum of square of two smaller numbers = square of third (larger) number a² + b² = c² 52 + 122 132 25 + 144 169 169 169 Ans:- Since sum of square of two small number is equal to square of third number therefore the numbers 5,13 and 12 form a Pythagorean Triplet. In 5, 13 and 12, the smaller numbers are 5 and 12. a= 5 and b = 12 and c=13 =

17. Example 3: Check whether the following can be the sides of a right angled triangle • AB = 6 cm, BC = 8 cm, AC = 10cm. Solution Pythagorean Triplet : Sum of square of two smaller numbers = square of third (larger) number a² + b² = c² Sum of square of two smaller sides (a²+b²)= 62 + 82 = 36 + 64 = 100 Square of third side (c² ) = 102 = 100 Ans:- Since sum of square of two smaller side is equal to square of third side therefore we can form right angle triangle from the given sides. In 6, 8 and 10, the smaller numbers are 6 and 8. a= 6 ,b = 8 and c=10

18. Example 4 : Check whether the following can be the sides of a right angled triangle • AB = 7 cm, BC = 10 cm, AC = 9 cm. • Solution: Pythagorean Triplet : Sum of square of two smaller numbers = square of third (larger) number • a² + b² = c² • Sum of square of two smaller sides (a²+b²) = 72 + 92 • = 49 + 81 • = 130 • Square of third side (c² ) = 102 • = 100 • Ans:- Since sum of square of two smaller side is not equal to square of third side therefore we cannot • form right angle triangle from the given sides. In 7, 10 and 9, the smaller numbers are 7 and 9. a= 7 and b = 9 and c=10

19. Try These 1. Can a right angled triangle have sides 12 cm, 14 cm and 15cm ? Find and fill the following table 2. Do the numbers 5, 6 and 7 form a Pythagorean Triplet? Find and fill the following table.