Enlargements

Last time

Before we start: last time, in Three-dimensional problems, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

The photocopier

A photocopier can enlarge a page. Every line gets longer by the same scale factor, and the whole picture grows away from, or towards, a fixed point. That fixed point is the centre of enlargement. Give the centre and the scale factor, and the image is fixed completely.

What an enlargement does

An enlargement keeps the shape but changes the size. All angles stay the same. Every length is multiplied by the scale factor k. With k equal to three, a side of four becomes twelve. The image is similar to the original, not congruent, unless k is one.

Three kinds of scale factor

If k is more than one, the image is bigger. If k is between zero and one, it is smaller. A fractional scale factor is still called an enlargement. A half shrinks a side of twelve to six. If k is negative, the image is on the opposite side of the centre and upside down.

Enlarging a point

With the centre at the origin, multiply both coordinates by k. The point two, three with k equal to three goes to six, nine. A negative scale factor flips it through the origin. With k equal to minus two, two, three goes to minus four, minus six. A fraction shrinks it: with k equal to a half, eight, four goes to four, two.