How can you picture the meaning of the Pythagorean Theorem? What are the geometric meanings of expressions, equations, and all of algebra? Keep on reading to find out!
The Geometric Meaning of Expressions
Let’s start by taking a closer look at all of those unit squares you’ve drawn. First, think about how many fit inside each larger square. For example, as you can see in your drawing, you can fit 4 unit squares from left-to-right across the square attached to the 4-unit-long leg of the triangle, and you can also fit 4 unit squares from top-to-bottom. So a total of 4 x 4 = 16 unit squares will fit inside this larger square. No big surprise!
But now imagine that this larger square doesn’t have a length and width of 4, but instead has a length and width represented by a variable called a. That means that the larger square measures a 1-by-1 unit squares across by a 1-by-1 unit squares high. The number of unit squares that will fit inside the larger square is therefore equal to a x a = a^2. Do you see what that means?
In the case of our 3–4–5 triangle whose sides are represented by the variables a, b, and c, it means that you can think of the a^2 part of the Pythagorean Theorem as the number of 1-by-1 unit boxes that will fit inside a 3-by-3 box, and the b^2 part of the equation as the number of 1-by-1 unit boxes that fit inside a 4-by-4 box. And what about c^2? Yep, you guessed it—it’s exactly the same idea but for the big box drawn along the hypotenuse of the triangle.