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Video: Applying Pythagoras's Theorem to Solve Complex Problems asd

Bethani Gasparine

Given that 𝐴𝐵𝐶𝐷 is a square, find the length of 𝐷𝐸.

03:40

Video Transcript

Given that ๐ด๐ต๐ถ๐ท is a square, find the length of ๐ท๐ธ. Since ๐ท๐ธ is what we are gonna be looking for, letโ€™s go ahead and call it ๐‘ฅ. Side ๐ท๐ธ is attached to the triangle. Letโ€™s go ahead and look at all the sides of the triangle.

We know that ๐ธ๐ถ or ๐ถ๐ธ is eighteen centimeters. And the side ๐ท๐ถ is not labeled. However, we do know what that is. Since ๐ด๐ต๐ถ๐ท is a square, that means all angles are ninety degrees and all sides are equal. Therefore, side ๐ท๐ถ is twenty-four centimeters.

One more important piece about this triangle is the angles. By angle ๐ถ in the square, thatโ€™s a ninety-degree angle. This would mean our triangle angle by angle ๐ถ would also be ninety degrees. Letโ€™s go ahead and redraw our triangle and see what we have and see if we can solve for ๐ท๐ธ.

We know side ๐ธ๐ถ is eighteen centimeters, side ๐ท๐ถ is twenty-four centimeters, and we wanna find ๐ท๐ธ. 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 ๐ด and ๐ต 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 ๐ท๐ธ๐ถ, twenty-four and eighteen are the legs, and ๐ท๐ธ, the side weโ€™re looking for, would be the hypotenuse, which we can replace with the variables ๐ด, ๐ต, and ๐ถ, where ๐ถ needs to be our hypotenuse, which is the side weโ€™re actually gonna be solving for ๐ท๐ธ. Letโ€™s go ahead and plug these in.

Using the Pythagorean theorem, letโ€™s substitute twenty-four centimetres in for ๐ด and eighteen centimeters in for ๐ต. Instead of putting centimeters in our equation, weโ€™ll just write it on our answer at the end. And our hypotenuse, instead of ๐ถ weโ€™re gonna be plugging in ๐‘ฅ. Thatโ€™ll be what ๐ท๐ธ will be. So letโ€™s 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 ๐‘ฅ, we get ๐‘ฅ 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 ๐‘ฅ, we need to square root both sides. And we get the square root of nine hundred is thirty and the square root of ๐‘ฅ squared is ๐‘ฅ. So if thirty is equal to ๐‘ฅ, then ๐ท๐ธ is equal to thirty centimeters. Again, ๐ท๐ธ is equal to thirty centimeters.