The Pythagorean Theorem is an important and essential topic to understand in mathematics. Without it, many mathematicians would be in the dark about the properties of triangles. The theorem is so important, it is useful in almost all the different types of school mathematics, like Algebra, Geometry, Trigonometry, and even Calculus. Understanding it will help you in all these different types of mathematics.
It is important to remember that Pythagorean's Theorem is only applicable in the case of right triangles. That means that it only works with triangles that have one right angle, or 90 degrees.
According to the theorem, if a triangle has three sides, labeled as a, b, and c, with c being the hypotenuse (the longest side) and a and b being the legs (the shorter sides), then: a squared + b squared = c squared. In other words, the sum of the squares of the shorter sides is equal to the square of the longer side. Sounds complicated, doesn't it? Well, maybe this will simplify things.
Imagine that you have a sheet of graph paper and a pencil. With the pencil, draw a line 3 cubes long, and then, connected from that line and going perpendicular from it, draw a line 4 cubes long. Now, connect the two lines together with a diagonal line (Just for the record, that diagonal line is what is called the hypotenuse, and the 3 and 4 lengthed sides are what's called the legs). Now, you know the length of two sides, but you do not know the length of that third side. In order to find it graphically, draw a square for each side of the triangle (except for the side you are trying to figure out), with the sides being the square's length and width. Add up the total cubes in both the squares. What would be the total? If you guessed 25, then you are right. If you were then to find the positive square root of that number, you would find the length of that third side, which is 5.
The same thing could've been done algebraically as well. Since 3 and 4 are your two legs, plug them into the formula as a and b. Then, use the formula to solve for c:
a squared + b squared = c squared.
(3) squared + (4) squared = c squared
9 + 16 = c squared
25 = c squared
c = 5
By using the formula, you were also able to find out the length of that final side, c. It came out to be 5.
Like I stated earlier, the Pythagorean Theorem is a fundamental topic to understand in mathematics. It can be used everywhere, and I hope that now since you understand it, you will.