JAMB 1986 · UME · Q33

Circle geometry

In the diagram, $PQ$ and $RS$ are chords of a circle with centre $O$ which, produced, meet at $T$ outside the circle. If $TP = 24$ cm, $TQ = 8$ cm and $TS = 12$ cm, find $TR$ .

Worked solution (try it first)

For two secants from an outside point $T$ : (outside part) × (whole line) is the same on both.
So $TQ \times TP = TS \times TR$ .
Put in the lengths: $8 \times 24 = 12 \times TR$ , so $192 = 12 \times TR$ .
Divide both sides by 12: $TR = 16$ cm, option A.

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