Discussion
In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and line QR. What is the ratio of the area of the triangle PST to the area of the trapezium SQRT?
Note: The figure shown is representative.
Answer: A
The ratio is \(\frac{1}{3}\).
Let the distance from P to QR be \(h\). The gap between ST and QR is \(h/2\), so ST is at a distance \(h/2\) from P, halfway down. Because ST is parallel to QR, triangle PST is similar to triangle PQR with ratio \(1:2\), and areas scale as the square of that ratio:
\[ \begin{aligned} \text{Area}(PST) &= \tfrac{1}{4}\,\text{Area}(PQR) \\ \text{Area}(SQRT) &= \tfrac{3}{4}\,\text{Area}(PQR) \\ \text{ratio} &= \frac{1/4}{3/4} = \frac{1}{3} \end{aligned} \]
\(\frac{1}{4}\) is the ratio of triangle PST to the whole triangle PQR, not to the trapezium, and \(\frac{1}{2}\) is the ratio of the lengths ST and QR, not of the areas.
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