Dilating triangles: find the error | Performing transformations | Geometry | Khan Academy

We are told triangle A'B'C' is the image of triangle ABC under a dilation whose center is P and scale factor is three-fourths. Which figure correctly shows triangle A'B'C' using the solid line? So pause this video and see if you can figure this out on your own.

All right, now before I even look at the choices, I like to just think about what that dilation actually looks like. So our center of dilation is P and it's a scale factor of (\frac{3}{4}). One way to think about it is however far any point was from P before, it's not going to be three-fourths as far, but along the same line.

So I'm just going to estimate it. If C was there, three half would be this way far, so three-fourths would be right about there. So C' should be about there. If we have this line connecting B and P like this, three... let's see, half of that is there. Three-fourths is going to be there, so B' should be there.

Then on this line, halfway is roughly there—I'm just eyeballing it—so three-fourths is there. So A' should be there, and so A'B'C' should look something like this, which we can see is exactly what we see for choice C. So choice C looks correct. I'm going to just circle that or select it just like that.

But let's just make sure we understand why these other two choices were not correct. So choice A, it looks like it is a dilation with a (\frac{3}{4}) scale factor. Each of the dimensions—each of the sides of this triangle—looks like it's about (\frac{3}{4}) of what it originally was, but it doesn't look like the center of dilation is P here. The center of dilation looks like it is probably the midpoint between or the midpoint of segment AC because now it looks like everything is (\frac{3}{4}) of the distance it was to that point.

So they have this other center of dilation in choice A. The center of dilation is not P, and that's why we can rule that one out.

Then for choice B, right over here, it looks like they just got the scale factor on. Actually, they got the center of dilation and the scale factor wrong. It still looks like they are using this as a center of dilation, but this scale factor looks like it's much closer to one-fourth or one-third, not three-fourths. So that's why we can rule that one out as well.

We like our choice C.