Scaling perimeter and area example 2
About this video
Quadrilateral A (perimeter 30, area 54) is dilated by a scale factor of 2/3. Perimeter scales by 2/3 to give 20, and area scales by (2/3)^2 = 4/9 to give 24. A 6-by-9 rectangle shrunk to 4-by-6 confirms the results.
You'll be able to
- scale a perimeter by a fractional scale factor.
- scale an area by the square of a fractional scale factor.
- check scaling results with a concrete example.
Key words
- dilate
- To enlarge or shrink a shape by multiplying every length by the same number.
- scale factor
- The number lengths are multiplied by; a scale factor less than 1, like 2/3, shrinks the shape.
- quadrilateral
- A flat shape with four straight sides.
Transcript
- [Presenter] 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.
Alright, so in previous videos we talk about if you have a scale factor, the 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 times 2/3. Let me write that a little bit neater.
So times 2/3 is going to be 20. And then the 54 is going to be scaled by 2/3 squared. One way to think about it, you're scaling in each dimension by 2/3, and so when you multiply the two dimensions to get area, you're gonna 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 times 4/9? That is equal to 54 times 4 over 9. Both 54 and 9 are divisible by 9. So let's divide them both by 9. This becomes a 6. This becomes a 1. And so we end up with 6 times 4 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... Let's say that that dimension is 6. That dimension is 9.
I think that adds up. 6 plus 6 is 12. 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, instead of having a length or a height of 6 over here, it's going to have a height of 4. And instead of having a length or a width of 9 here, it's gonna be 2/3 of that. It's going to be 6. And so the quadrilateral will look like this. And we can verify that the perimeter now is gonna be 4 plus 4, which is 8 plus 6 plus 6, 12.
So it's 8 plus 12, which is 20. And the new area is 6 times 4 is 24. Now, you didn't have to do this, but I just wanted to make sure you understood why this was happening.
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