Thales' theorem: when triangles zoom

The fact to remember

If two intersecting lines are cut by two parallel lines, the lengths on the two lines are in the same ratio.

The unlock example

You want to measure a tree’s height without climbing it.

• Your stick is 1 m, its shadow is 1.5 m.
• The tree’s shadow is 12 m.

Sunlight makes parallel rays → Thales applies. The tree’s height follows the same ratio:

$\frac{\text{tree height}}{\text{tree shadow}} = \frac{\text{stick height}}{\text{stick shadow}} \Rightarrow \frac{h}{12} = \frac{1}{1.5}$

So $h = \frac{12}{1.5} = 8$ meters.

The classic mistake to avoid

Always check that the lines really are parallel before applying Thales. Two segments that « look parallel » aren’t enough.

?Your turn

On a Thales figure, AB = 4, AB' = 10 and AC = 6. What is AC'?

A.12B.15C.10D.24

Check