Definition and ratio of a homothety

What is a homothety?

A homothety is a transformation of the plane that enlarges or reduces a figure from a fixed point called the centre, by a factor called the ratio (often denoted by k).

We denote the homothety with centre O and ratio k by h(O, k). If M’ is the image of M, then the points O, M and M’ lie on a straight line, and the following vector relationship holds: vector OM’ = k × vector OM.

Interpreting the ratio k

The ratio k provides two key pieces of information:

Value of k	Effect on the figure	Position of M’ relative to O
k > 1	Enlargement	M’ on the same side as M, further from O
k = 1	No change (identity)	M’ = M
0 < k < 1	Reduction	M’ on the same side as M, closer to O
k = -1	Central symmetry	M’ on the opposite side of O, OM’ = OM
k < 0	Reflection + change in size	M’ on the opposite side of O

Example

Let O be a point and M a point such that OM = 4 cm. We apply a homothety with centre O and ratio k = 2.5.

We then have OM' = 2.5 × 4 = 10 cm, with M' collinear with O and M, on the same side (since k is positive).

If we had chosen k = -0.5, we would have OM’ = 0.5 × 4 = 2 cm, but M’ would lie on the opposite side of O relative to M (since k is negative).

Common pitfall

Don’t forget the sign of k! A negative ratio not only changes the size, it also changes the direction: point M’ moves to the other side of the centre O. Many pupils overlook this reversal and place M’ on the wrong side.