Question

Two right triangles $PRQ$ and $PSQ$ are drawn on the same hypotenuse $PQ$ . If $PR$ and $QS$ intersect at $T$ , prove that $ST \times TQ = PT \times TR$ .

Approach

Since $△ STP \sim △ RTQ$ (by AA similarity via right angles and common angles), corresponding sides are in proportion: $\frac{ST}{TP} = \frac{RT}{TQ}$ , which cross-multiplies to $ST \times TQ = PT \times TR$ .

Step-by-step working

1
Identify $△ STP \sim △ RTQ$
1 mark
2
Use proportionality $\frac{ST}{TP} = \frac{RT}{TQ} \Rightarrow ST \times TQ = PT \times TR$
1 mark

Concepts used

similar trianglesright trianglesproportionalityproof

Approach

Step-by-step working

Concepts used

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