The lengths of the sides of the right triangle shown in the figure are three, four, and five. Find the areas of the squares on the three sides, and find a relationship between them.
So we have a right triangle in the middle. And it says that the sides of this right triangle are three, four, and five. And looking at the tiny boxes, we can see this side must be the length of three because of the one, two, three boxes. This will be four. And this last one, the hypotenuse, will be five. It says to find the areas of the squares.
Let’s begin with this small square. We could count all of the spaces, the blocks. And they’re nine. Or we could say this is a three-by-three square. So the length and the width are each three. And to find the area, so we would take length times width to be three times three, which is nine, just like we found.
Now, let’s move to the other square on the other leg. We can either count each of the tiny squares. And that would be 16. Or this is a four-by-four square, so length times width. And four times four would indeed give us 16. Lastly, we have the largest square, the square on the hypotenuse. We could count each of the boxes, the tiny boxes, and get 25 or take five times five, the length times the width.
So we found the areas of the squares on the three sides. And now we need to find a relationship between them. We have nine, 16, and 25. Now notice, nine and 16 add together to equal 25. So we could say that the area of the square on the hypotenuse, which is 25, is equal to the sum of the areas of the squares on the legs, 16 plus nine. 16 plus nine is equal to 25.
Well, this is a perfectly fine answer. How did we get here? What exactly are we describing? Well, we’re working with the right triangle. And we’ve stated that the square on the hypotenuse is equal to the sum of the areas of the squares on the legs.
So the square on the hypotenuse — how was that made? Well, it was made from taking five times five, the area of the square. Well, five times five is the same thing as five squared. And for nine, instead of three times three, we could say three squared. And for 16, instead of four times four, we could say four squared. So the square of the hypotenuse is equal to the sum of the squares on the legs. Five squared is equal to three squared plus four squared. Five squared is 25. Three squared is nine. Four squared is 16. And nine plus 16 is equal to 25.
So what theorem is this? This is a theorem that we’re describing that can be used with right triangles, the Pythagorean theorem. So the relationship that we described was a Pythagorean theorem.
So once again, our relationship between the areas of the squares on these three sides would be the area of the square on the hypotenuse, 25, is equal to the sum of the areas of the squares on the legs, 16 plus nine.