The *Pythagorean Theorem Diagrams* is used to find the length of one of the sides of a right triangle knowing its other two sides. This theorem has been of great aid to mankind in many areas. In the area of architecture, this theorem is used in building design and construction, for example to calculate the length of a right-triangular truss.

*Pythagorean theorem* and its proof are closely related to the areas of a square and triangle. Therefore, to understand Pythagorean theorem you have to have knowledge of finding the areas of a square and triangle. To refresh your memory on the formulas for the area of a **square** and a **right triangle, **see the following description!

1. The area of ABCD square = AB x BC = AB x AB (since AB = BC) = AB². So the **area of square = side length x side length.**

2. The area of a **Right Triangle ABC =** 1/2 x AB x AC.

If AB and AC are the legs of the triangle, then the **area of the right triangle = 1/2 x length of leg I x length of leg II**

A right triangle has two *legs* or *catheti* (singular:cathetus) and *hypotenuse. *The first figure above is a right triangle ABC having a right angle at the vertex A. The two sides that meet at a right angle, namely AB and AC are called its **legs**. The side opposite to the right angle, namely BC is called its **hypotenuse**.

**BC² = AC² + AB² **,or

**a² = b² + c²**

Look at the diagram above!

Figure (i) and (ii) above show two squares whose *sides have equal length* of (b+c). Having *equal side length*, the *areas of both squares must also be equal*. The *shaded* region in figure i and ii are *equal in area*. This in turn shows that the *non-shaded* regions are also *equal in area*. Hence, **a² = b² + c².**

Next look at the figure (iii). The figure is constructed from rearranging pieces in figure i and ii. The area of the square on the hypotenuse is a² and the sum of the areas of the squares on the legs is (b² + c²).

From the discussion above, for every **right triangle** = the **area of the square** on the **hypotenuse** equals the** sum of the areas of the squares** on the other two sides (**legs**)

Let’s practice and find answer the question in picture above!

**Answer:**

p² = 15² – 12²

p² = 255 – 144

p² = 81

p = √81 = 9

Thus, **p = 9**