Similarity & Transformations

The scale factor

The scale factor compares an image side to the matching original side:

$k = \dfrac{\text{image side}}{\text{original side}}$

Every side of the image equals the matching original side multiplied by $k$. If $k$ is greater than $1$, the image is an enlargement. If $k$ is between $0$ and $1$, the image is a reduction. And if $k = 1$, the image is the same size — so the figures are not just similar, they're also congruent.

In the figure below, the larger triangle is a dilation of the smaller one with $k = 2$: every side of the image is exactly twice the matching original side.

Finding k from a picture

Pick one side of the original figure whose length you know, find the matching side on the image, and divide image by original. That single division gives you $k$ for the entire figure — you never need to check more than one pair, because a dilation stretches every side by the same factor.

Direction matters. Dividing image by original tells you how to get from the original to the image. If you accidentally divide original by image, you'll get the scale factor for going the other way — the reciprocal of the $k$ you wanted.

Using k to find missing sides

Once you know $k$, multiply. Each side of the image is the matching original side times $k$ — a dilation never adds a fixed amount to each side. A triangle with sides $5$, $7$, and $10$ dilated by $k = 2$ becomes $10$, $14$, and $20$, not $7$, $9$, and $12$.

Angles don't change at all. A dilation keeps every angle the same measure, which is exactly why the shape survives while the size changes.

Worked examples

Try one yourself

Common questions