Given that 𝐴𝐵𝐶𝐷 is a square, find the length of 𝐷𝐸. Since 𝐷𝐸 is what we are gonna be looking for, let’s go ahead and call it 𝑥. Side 𝐷𝐸 is attached to the triangle. Let’s go ahead and look at all the sides of the triangle.
We know that 𝐸𝐶 or 𝐶𝐸 is eighteen centimeters. And the side 𝐷𝐶 is not labeled. However, we do know what that is. Since 𝐴𝐵𝐶𝐷 is a square, that means all angles are ninety degrees and all sides are equal. Therefore, side 𝐷𝐶 is twenty-four centimeters.
One more important piece about this triangle is the angles. By angle 𝐶 in the square, that’s a ninety-degree angle. This would mean our triangle angle by angle 𝐶 would also be ninety degrees. Let’s go ahead and redraw our triangle and see what we have and see if we can solve for 𝐷𝐸.
We know side 𝐸𝐶 is eighteen centimeters, side 𝐷𝐶 is twenty-four centimeters, and we wanna find 𝐷𝐸. Since we have a right triangle, we are gonna be able to use the Pythagorean theorem. The Pythagorean theorem describes the relationship between the lengths of the legs and hypotenuse for any right triangle. Looking at our diagram, you can see that 𝐴 and 𝐵 are the legs, and the longest side, the one across from the ninety-degree angle, is the hypotenuse.
In any right triangle, the sum of the squares of the lengths of the legs are equal to the length of the hypotenuse squared. So looking at our triangle 𝐷𝐸𝐶, twenty-four and eighteen are the legs, and 𝐷𝐸, the side we’re looking for, would be the hypotenuse, which we can replace with the variables 𝐴, 𝐵, and 𝐶, where 𝐶 needs to be our hypotenuse, which is the side we’re actually gonna be solving for 𝐷𝐸. Let’s go ahead and plug these in.
Using the Pythagorean theorem, let’s substitute twenty-four centimetres in for 𝐴 and eighteen centimeters in for 𝐵. Instead of putting centimeters in our equation, we’ll just write it on our answer at the end. And our hypotenuse, instead of 𝐶 we’re gonna be plugging in 𝑥. That’ll be what 𝐷𝐸 will be. So let’s evaluate twenty-four squared and eighteen squared.
We have five hundred and seventy-six plus three hundred and twenty-four equals — when we square 𝑥, we get 𝑥 squared. Now we need to add our two numbers on the left-hand side of the equation to get nine hundred.
In order to solve for 𝑥, we need to square root both sides. And we get the square root of nine hundred is thirty and the square root of 𝑥 squared is 𝑥. So if thirty is equal to 𝑥, then 𝐷𝐸 is equal to thirty centimeters. Again, 𝐷𝐸 is equal to thirty centimeters.