Properties of homothety

What a homothety preserves

Unlike rotation, a homothety does NOT preserve lengths (unless k = 1 or k = -1). However, it does preserve:

• angles
• parallelism (a line and its image are parallel)
• alignments
• the nature of the shapes (a square remains a square, a circle remains a circle)

Effect on lengths and areas

If an initial length is L, its image is |k| × L (the absolute value of k, as a length is always positive).

For areas, the effect is even more pronounced: an initial area A becomes k² × A after the homothety. This is logical, as an area depends on the product of two lengths, each multiplied by k.

Example

A triangle has an area of 6 cm². A homothety with ratio k = 3 is applied to it.

• The lengths of the sides are multiplied by 3
• The area of the image triangle is 3² × 6 = 9 × 6 = 54 cm²

If the ratio had been k = -2, the lengths would have been multiplied by 2 (absolute value), and the area by (-2)² = 4, i.e. 24 cm², even though the triangle is flipped about its centre.

Homothety and circles

The image of a circle with centre A and radius r under a homothety with ratio k is a circle with radius |k| × r, whose centre is the image of A.

Common pitfall

Do not confuse the effect on lengths (multiplied by |k|) with the effect on areas (multiplied by k²). For example, when k = 2, the lengths double but the area is multiplied by 4, not by 2: this is the most common mistake in this chapter.