Solving aspect ratio problems with the Pythagorean theorem

0:00- [Instructor] Elena is designing a rectangular whiteboard with a 3:4 aspect ratio. The diagonal of the whiteboard measures 60 inches. What is the area of the whiteboard? Pause this video, have a go at that. All right. So when they say it's a 3:4 aspect ratio, that means that the ratio between the width and the height is 3:4.

0:22So for example, let's say that this is three x, whatever x is, then this would be four x. Now what we can do is use this information and the fact that it's a rectangular whiteboard, which means that this is a right angle. We can use this in the Pythagorean theorem to actually figure out what x is.

0:42And then once we know what x is, we can figure out what 3x is and 4x is and figure out the area of the whiteboard. So how do we use a Pythagorean theorem here? Well, the Pythagorean theorem says the sum of the squares of the two shorter sides is equal to the length of the hypotenuse squared.

0:58So this would be four x squared plus three x squared. The sum of the squares of the two shorter sides is equal to the square of the hypotenuse or the length of the hypotenuse. And the hypotenuse is the longest side. It's the side that is opposite the right angle.

1:18And so let's see if we could figure out what x is. This 4 x squared can be the same or it is the same thing as 16 x squared. And then 3 x squared is the same thing as 9 x squared. And this is equal to 60 times 60 is 3,600. And see, 16 of something plus nine of something, and both of these are 16 x squared plus 9 x squared is 25 x squared, is equal to 3,600.

1:49To solve for x squared, we could divide both sides by 25, divide both sides by 25. And we are going to get x squared is equal to 3,600 over 25. Now I could figure out what 3,600 over 25 is. I think that'd be the same thing as four times 36. Or I could just take the square root of both sides because we have a perfect square in the numerator and a perfect square in the denominator.

2:15So x is going to be equal to, you could say if we didn't know this was a geometry problem and if you knew that x had to be positive, you would say it's the plus or minus square root of the right. But we know it's got to be the positive square root because we're not dealing with negative dimensions.

2:28So if we take the positive square root of 3,600 over 25, that's going to be equal to 60 over five, which is equal to 12. But we are not done. They did not ask us for x. They asked us for the area of the whiteboard. So three x over here is going to be equal to three times 12, which is equal to 36.

2:54And four x is equal to four times 12, which is equal to 48. So the area is going to be the base times the height, which is 36 times 48. Or I could write 48 times 36. Do a little bit of math here. Let's see. Eight times six is 48. Four times six is 24 plus four is 28.

3:19Now I'm going to put the zero because I'm multiplying by the 10s place. Three times eight is 24. And then three times four is 12 plus two is 14. And we get eight, eight plus four is 12. Regroup that one, one plus two plus four is seven. And then we have this one here.

3:40So what's the area of the whiteboard? 1,728 square inches. And we are done.