Video Transcript
The image of a shape has a perimeter of 40 following a dilation by a scale factor of one-half. What would the perimeter of the original shape be?
In this question, we’re given the information about the image of a shape following a transformation. And that transformation is a dilation. We’re told that this image has a perimeter of 40 and the dilation had a scale factor of one-half. And we need to work out the perimeter of the original shape. We might guess that if the image has a perimeter of 40 after dilation by a scale factor of one-half, then the original perimeter must be 80 units. But it’s worth having a little investigation to check.
We’re not told what the shape is. For example, it could be a rectangle or it could be a triangle. It could be absolutely anything, even a hexagon or an irregular shape. But let’s look at these two shapes. When there’s a dilation with a scale factor of one-half, then every length on the image will be half the length on its original shape. So we have the original and the image of the rectangle and the triangle. Let’s label the lengths on our rectangle as 𝑥 and 𝑦, and we’re thinking about the perimeter.
We remember that the perimeter is the distance around the outside of the shape. So for the larger rectangle, we’ll have 𝑥 plus 𝑦 plus 𝑥 plus 𝑦, which would simplify to two 𝑥 plus two 𝑦. We could even take out the common factor of two and write it as two and then 𝑥 plus 𝑦 in parentheses. If we then think about the lengths on our shape after the dilation, then these would of course be half 𝑥 and half 𝑦. The perimeter here would be half 𝑥 plus half 𝑦 plus a half 𝑥 plus a half 𝑦. This would simplify to 𝑥 plus 𝑦.
If we now compare the two perimeters on the image and the original shape, we have two times 𝑥 plus 𝑦 and 𝑥 plus 𝑦. This means that the image would have a perimeter that also has a scale factor of one-half. However, does this also work with the triangles? This time, we can say that the lengths on the triangle are 𝑎, 𝑏, and 𝑐. After a dilation with a scale factor of one-half, then the lengths on the image would be one-half 𝑎, one-half 𝑏, and one-half 𝑐. The perimeter on the original shape would be 𝑎 plus 𝑏 plus 𝑐. The perimeter of the image would be a half 𝑎 plus a half 𝑏 plus a half 𝑐.
We can factor out the common factor of one-half to give us a half times 𝑎 plus 𝑏 plus 𝑐. If we compare the perimeters of the original shape and the image, then once again we can see that the perimeter would have a scale factor of one-half. Although we’ve just taken two instances here of a rectangle and a triangle, we can see how whenever we transform a shape with a dilation of a scale factor of one-half, then the perimeter will also have a scale factor of one-half. So, that means for our question that when the perimeter of the image of a shape is 40 after a dilation of one-half, then we would indeed get a perimeter of 80. And so the answer is 80 length units.
It’s important to note that the same is not true with area. If we have a dilation by a scale factor of one-half, then the area would not have a scale factor of one-half, just the perimeter.
