**Pythagorean Theorem**, states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

The theorem can be written as an equation relating the lengths of the sides *a*, *b* and *c*, often called the "Pythagorean equation"

where *c* represents the length of the hypotenuse and *a* and *b* the lengths of the triangle's other two sides.

If *c* denotes the length of the hypotenuse and *a* and *b* denotes the lengths of the other two sides, the Pythagorean theorem can be expressed as the Pythagorean equation:

If the length of both *a* and *b* are known, then *c* can be calculated as

If the length of the hypotenuse *c* and of one side (*a* or *b*) are known, then the length of the other side can be calculated as

or

The Pythagorean equation relates the sides of a right triangle in a simple way, so that if the lengths of any two sides are known the length of the third side can be found.

**Corollary of the theorem: **In any right triangle, the hypotenuse is greater than any one of the other sides, but less than their sum.

A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation.

*Pythagorean Theorem can be used **to* solve real world problems in Architecture and Construction, Laying Out Square Angles, Navigation, Surveying.

**Demonstration**

Let’s consider the Height of the tree to be 12 feet tall, then id we can place ladder at 6 feet away from the trees base then we can arrive at the following solution for finding length of ladder needed.

a = 12

b = 6

c= length of ladder

c= √ 144+36

= 13.41= 13 feet