JAMB 1984 · UME · Q44

In the figure, $\angle TSP = \angle PRQ$ , $QR = 8$ cm, $PR = 6$ cm and $ST = 12$ cm. Find the length $SP$ .

Worked solution (try it first)

Triangles $PRQ$ and $PST$ share $\angle P$ and have $\angle PRQ = \angle PST$ , so they are similar, with $R$ matching $S$ and $Q$ matching $T$ .
Corresponding sides are in the same ratio: $\dfrac{PR}{PS} = \dfrac{QR}{TS}$ , so $\dfrac{6}{SP} = \dfrac{8}{12}$ .
Cross-multiply: $8 \times SP = 72$ , so $SP = 9$ cm, option C.