GATE CS 2025 Set 1 — Question 7
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General Aptitude → Quantitative Aptitude → Mensuration & Geometry
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Question
In the diagram, the lines and are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point and line . What is the ratio of the area of the triangle to the area of the trapezium ?Note: The figure shown is representative.
Correct answer
(A) (1)/(3)
Solution
Let be the height of from vertex to the base . This is the shortest distance between and line .
Let be the shortest distance between the parallel lines and . According to the problem, .
The height of the smaller triangle from vertex to base is .
Since , is similar to ().
The ratio of their corresponding heights is .
The ratio of their areas is the square of the ratio of their corresponding linear dimensions:
.
Let . Then .
The area of the trapezium is .
The required ratio is .
Let be the shortest distance between the parallel lines and . According to the problem, .
The height of the smaller triangle from vertex to base is .
Since , is similar to ().
The ratio of their corresponding heights is .
The ratio of their areas is the square of the ratio of their corresponding linear dimensions:
.
Let . Then .
The area of the trapezium is .
The required ratio is .
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