# Video: Solving for the Hypotenuse of a Right-Angled Triangle with Noninteger Solutions

Which of the relations below is correct? [A] 7² = 𝑥² − 16 [B] 𝑥 = 4² + 7² [C] 𝑥² = (7 + 4)² [D] 49 = 𝑥² + 16.

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### Video Transcript

Which of the relations below is correct? Seven squared equals 𝑥 squared minus 16. 𝑥 equals four squared plus seven squared. 𝑥 squared equals seven plus four squared. Or, 49 equals 𝑥 squared plus 16.

So we are given a right triangle. And we know two of the side lengths, four centimeters and seven centimeters. The third side, which is the unknown side, we call 𝑥. And since it’s the side across from the 90-degree angle, this is the longest side. And this is important in order to use the Pythagorean theorem. The Pythagorean theorem states: the square of the longest side is equal to the sum of the squares of the shorter sides. So we already know that the longest side is represented by 𝑥. So we can plug that in. Now the other two sides are the shorter sides. And they’re seven and four. And now we can plug these in. So we have 𝑥 squared equals seven squared plus four squared.

Let’s begin to look at our options for answers. Option B is very close, except 𝑥 needs to be squared. And the four squared and seven squared are in different spots. But that’s okay. When adding, we could have switched them around. It wouldn’t change anything. But again, that 𝑥 needed to be squared. So we can eliminate B.

C looks very close as well. 𝑥 squared equals seven plus four, then squaring. So is seven squared plus four squared the same as seven plus four squared? Let’s check. So for seven plus four squared, we need to add first. So seven plus four is 11. And 11 squared is 121. So now, let’s check the other, seven squared plus four squared. Seven squared is 49. And four squared is 16. And 49 plus 16 is 65. So these are not equal. Therefore, we can eliminate option C.

Let’s compare options A and D because they’re very similar. Option A is: seven squared equals 𝑥 squared minus 16. And option D is: 49 equals 𝑥 squared plus 16. So the sign on the 16s are different. And one of them is equal to seven squared. And one of them is equal to 49. However, seven squared is equal to 49. So those actually mean the same thing. So the only difference is, one is 𝑥 squared minus 16 and one is 𝑥 squared plus 16.

So let’s go back to our equation, created from the Pythagorean theorem and manipulated so the 𝑥 squared and the 16 are on the same side, and see if we need a plus 16 or a minus 16. And the 16 will come from the four squared because four squared is 16. So in order to solve, let’s go ahead and subtract 16 from both sides. This way it’s on the same side as the 𝑥 squared. We have 𝑥 squared minus 16 equals seven squared, which is the same as seven squared equals 𝑥 squared minus 16 because it wouldn’t matter if the sides of the 𝑥 squared minus 16 and the seven squared were on opposite sides as long as their sign could stay the same.

Therefore, our option A is our correct answer: seven squared equals 𝑥 squared minus 16.