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)
Explanation
Let the distance from P to QR be \(h\). ST is \(\frac{h}{2}\) from QR, so it is also \(\frac{h}{2}\) from P. Because ST is parallel to QR, triangle PST is similar to triangle PQR with scale factor \(\frac{1}{2}\), so its area is \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\) of the area of PQR. The trapezium SQRT is the remaining \(\frac{3}{4}\), so the required ratio is \(\frac{1/4}{3/4} = \frac{1}{3}\).
Answer: A