6.3 Criteria for Similarity of Triangles

Let $AB$ be the pole and $BC$ be its shadow. $AB = 6, BC = 4$.

Let $PQ$ be the tower and $QR$ be its shadow. $QR = 28$. Find $PQ$.

At the same time of day, the sun's elevation angle is identical for both objects.

So, $\angle C = \angle R$ (Sun's angle) and $\angle B = \angle Q = 90^\circ$ (Vertical objects).

By AA Similarity, $\Delta ABC \sim \Delta PQR$.

Ratio of sides: $\frac{AB}{PQ} = \frac{BC}{QR}$.

Substitute: $\frac{6}{h} = \frac{4}{28}$.

Solve: $h = \frac{6 \times 28}{4} = 6 \times 7 = 42$ m.