Pythagoras Theoram

In a right angled triangle, the area of the square on hypotenuse is equal to the sum of the areas of squares on the remaining sides.

Pythagorean Triplet.

If a, b, c are natural numbers and a>b, then [(a2+b2), (a2-b2), (2ab)] is Pythagorean triplet. (a2+b2)2 = (a2)2 + 2a2b2 + (b2)2 (a2+b2)2 = a4+ 2a2b2 + b4   ............................(I) (a2 - b2)2= (a2)2 - 2a2b2 + (b2)2   (a2- b2)2 = a4 - 2a2b2 + b4  ............................(II) Subtracting (II) from (I), we get (a2+b2)2 - (a2- b2)2 = 4a2b2 (a2+b2)2 - (a2- b2)2 = (2ab)2 (a2+b2)2 = (a2- b2)2+ (2ab)2 Therefore [(a2+b2) , (a2- b2) and (2ab)] is Pythagorean Triplet. Taking different values of a & b we can find more and more triplets.
Property of 300-600-900 Triangle If acute angles of a right angled triangle are 300,600 and 900, then the side opposite 300 angle is half of the hypotenuse and the side opposite to 600 angle is times the hypotenuse.

Property of 30-60-90 type triangle.

Property of 450-450-90triangle If the acute angles of right angled triangles are 450 each, then each of the perpendicular side is times the hypotenuse.