### Video Transcript

Given that ๐ด๐ต๐ถ๐ท is a square, find the length of ๐ท๐ธ. Since ๐ท๐ธ is what we are gonna be looking for, letโs go ahead and call it ๐ฅ. Side ๐ท๐ธ is attached to the triangle. Letโs go ahead and look at all the sides of the triangle.

We know that ๐ธ๐ถ or ๐ถ๐ธ is eighteen centimeters. And the side ๐ท๐ถ is not labeled. However, we do know what that is. Since ๐ด๐ต๐ถ๐ท is a square, that means all angles are ninety degrees and all sides are equal. Therefore, side ๐ท๐ถ is twenty-four centimeters.

One more important piece about this triangle is the angles. By angle ๐ถ in the square, thatโs a ninety-degree angle. This would mean our triangle angle by angle ๐ถ would also be ninety degrees. Letโs go ahead and redraw our triangle and see what we have and see if we can solve for ๐ท๐ธ.

We know side ๐ธ๐ถ is eighteen centimeters, side ๐ท๐ถ is twenty-four centimeters, and we wanna find ๐ท๐ธ. 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 ๐ด and ๐ต 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 ๐ท๐ธ๐ถ, twenty-four and eighteen are the legs, and ๐ท๐ธ, the side weโre looking for, would be the hypotenuse, which we can replace with the variables ๐ด, ๐ต, and ๐ถ, where ๐ถ needs to be our hypotenuse, which is the side weโre actually gonna be solving for ๐ท๐ธ. Letโs go ahead and plug these in.

Using the Pythagorean theorem, letโs substitute twenty-four centimetres in for ๐ด and eighteen centimeters in for ๐ต. Instead of putting centimeters in our equation, weโll just write it on our answer at the end. And our hypotenuse, instead of ๐ถ weโre gonna be plugging in ๐ฅ. Thatโll be what ๐ท๐ธ will be. So letโs evaluate twenty-four squared and eighteen squared.

We have five hundred and seventy-six plus three hundred and twenty-four equals โ when we square ๐ฅ, we get ๐ฅ 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 ๐ฅ, we need to square root both sides. And we get the square root of nine hundred is thirty and the square root of ๐ฅ squared is ๐ฅ. So if thirty is equal to ๐ฅ, then ๐ท๐ธ is equal to thirty centimeters. Again, ๐ท๐ธ is equal to thirty centimeters.