Scaling perimeter and area example 2 | Transformational geometry | Grade 8 (TX) | Khan Academy

We're told quadrilateral A was dilated by a scale factor of 2/3 to create quadrilateral B. Complete the missing measurements in the table below. So like always, pause this video and then we will do this together. Try to do it yourself, and then we'll do it together.

All right, so in previous videos, we talked about if you have a scale factor, perimeter is going to be scaled by the same amount, while area is going to be scaled by the square of that. So perimeter is also going to be scaled by 2/3. So 30 * 2/3… let me write that a little bit neater. So times 2/3 is going to be 20.

Then the 54 is going to be scaled by (2/3) squared. One way to think about it is, you're scaling in each dimension by 2/3, and so when you multiply the two dimensions to get area, you're going to be multiplying by 2/3 twice to get the new scaled area. So what is (2/3) squared? Well, that is the same thing as 4/9. So what is 54 * (4/9)?

That is equal to 54 * (4 over 9). Both 54 and 9 are divisible by 9, so let's divide them both by 9. This becomes 6, and this becomes 1. So we end up with 6 * 4, which is equal to 24, and we're done.

Now to make this very tangible in your head, let's give an example of where this could actually happen. Let's imagine that quadrilateral A, let's say it looked like this, and I think I can eyeball it. Let's say that that dimension is 6, and that dimension is 9. I think that adds up: 6 plus 6 is 12, and 9 plus 9 is 18. So yes, this perimeter is 30, and the area here is actually 54.

So this is actually the example of quadrilateral A over here. And now quadrilateral B, if we're scaling it by 2/3, then all of these dimensions are going to be scaled by 2/3. So quadrilateral B will, instead of having a length of—or height of—6 over here, it's going to have a height of 4, and instead of having a length or width of 9 here, it's going to be 2/3 of that. It's going to be 6.

So the quadrilateral will look like this, and we can verify that the perimeter now is going to be 4 plus 4, which is 8, plus 6, plus 6, which is 12. So it's 8 plus 12, which is 20, and the new area is 6 * 4, which is 24. Now, you didn't have to do this, but I just wanted to make sure you understood why this was happening.