In the figure, given that the two triangles are similar, what is the scale factor that would take you from the larger triangle to the smaller triangle?
We can recall that the word “similar” means the same shape but a different size. More specifically, we can say that corresponding angles are congruent and corresponding sides are in proportion. In order to find the scale factor that takes us from the large triangle to the small triangle, we need to work out the proportion of the sides. This ratio or proportion is the scale factor.
We can start by looking at the base lengths and using the helpful formula that the scale factor is equal to the new length over the original length. Then, as the new length in the smaller triangle is six and the original base length is 12 in the larger triangle, we have a scale factor of six over 12, which simplifies to one-half. So, each of the lengths in the smaller triangle will be half of the lengths in the larger triangle.
We can check our answer using the other pair of given sides. If we take the length of 11 and multiply it by the scale factor of a half, we would get 11 over two, which simplifies to five and a half or 5.5. The corresponding length on the smaller triangle was indeed 5.5. And so, we’ve confirmed our answer that the scale factor from the larger triangle to the smaller triangle is a half.
We must always make sure that the scale factor is a multiplier. So, for example, we could’ve divided the lengths on the larger triangle by two to get to the smaller triangle. But our scale factor could not be “dividing by two.” It would have to be “multiplying by a half.”