Calculating a length using Thales’ theorem

The statement of the theorem

In a triangle ABC, if M is a point on [AB], N is a point on [AC], and (MN) is parallel to (BC), then:

AM/AB = AN/AC = MN/BC

These three ratios are equal: this is the key to calculating an unknown length.

Step-by-step example

Given that AB = 8 cm, AM = 5 cm and AC = 6 cm, we want to calculate AN.

According to Thales’ theorem: AM/AB = AN/AC, so 5/8 = AN/6.

We isolate AN: AN = 6 × 5/8 = 3.75 cm.

In the same way, if BC = 9 cm, we calculate MN: MN/BC = AM/AB, so MN = 9 × 5/8 = 5.625 cm.

Summary table for the example

Length	Value
AB	8 cm
AM	5 cm
AC	6 cm
AN	3.75 cm
BC	9 cm
MN	5.625 cm

Common pitfall: matching the points correctly

The most common mistake is to reverse the correspondences, for example writing AM/AB = AC/AN instead of AM/AB = AN/AC. To avoid this mistake, always write the lengths from the common vertex in the same order: first the shorter sides (AM, AN, MN), then the longer ones (AB, AC, BC).

Tip: draw a diagram, label the letters, and check that vertex A is in the same position in each ratio before calculating.