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. A. 1/3 B. 1/4 C. 2/5 D. 1/2

GATE 2025 · General Aptitude · Area · medium

Answer: A. The ratio of area(PST) to area(SQRT) is 1/3.

1. Find the similarity ratio of PST to PQR: ST is at distance h/2 from P and distance h/2 from QR. The height of triangle PST is h/2 while the total height is h, giving similarity ratio 1:2.
2. Compute area ratio of PST to PQR: Area scales as the square of the linear scale factor.
3. Find area of trapezium SQRT: The trapezium is the region between ST and QR, i.e. the larger triangle minus the smaller.
4. Compute the required ratio: Divide area(PST) by area(SQRT).