In the diagram, the lines QR and ST are parallel to each other. The shortest…
2025
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.
\(1\over 3\) - B.
\(1\over 4\) - C.
\(2 \over 5\) - D.
\(1\over 2\)
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Correct answer: A
Key idea: use similarity of triangles and area scaling.
Let the base QR have length B. Let the shortest distance from P to QR be 2d, so the distance between the lines ST and QR is d. Then the distance from P to ST is 2d − d = d.
Triangles PST and PQR are similar because ST is parallel to QR. The linear scale factor is (height from P to ST)/(height from P to QR) = d/(2d) = 1/2, so ST = (1/2)·QR.
Areas scale as the square of the linear factor, so area(PST) = (1/2)^2 · area(PQR) = 1/4 · area(PQR).
The trapezium SQRT is the part of triangle PQR below ST, so area(SQRT) = area(PQR) − area(PST) = (1 − 1/4)·area(PQR) = 3/4·area(PQR).
Therefore the ratio area(PST) : area(SQRT) = (1/4) : (3/4) = 1 : 3 = 1/3.
Answer: 1/3
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