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NCERT Solutions Ganita Prakash (Part 2) Chapter 7 Triangles between Parallel Lines with a Common Base — In-text Questions

Book page 156 Updated on2026-09-05

Q1.
(i) Which of these triangles has the maximum area, and which has the minimum area?
Answer

None of them — every one of these triangles has exactly the same area. There is no maximum and no minimum.

All the triangles share the base BC, and every third vertex lies on the line l, which is parallel to BC. The distance between two parallel lines is the same everywhere, so every one of these triangles has the same height.

height = distance between l and BC = d (the same for every position of the apex)
Area = ½ × BC × d, whatever the apex
All the areas are equal.
Why it happens: the apex sliding along l stretches the triangle sideways. Its two slanted sides get longer, but the base and the height — the only two numbers in the formula — never change. This is the single most useful fact in the chapter: triangles on the same base and between the same parallels are equal in area.
Q2.
(ii) Which of these triangles has the maximum perimeter, and which has the minimum perimeter?
Answer

There is no maximum perimeter, but there is a definite minimum.

  • No maximum: push the apex A further and further along l and the two slanted sides AB and AC grow without limit, so the perimeter grows without limit.
  • Minimum: BC is common to every triangle, so we only have to make AB + AC as small as possible. Reflect C in the line l to get C′. Then AC = AC′, so AB + AC = AB + AC′, which is a path from B to C′ through A. The shortest such path is the straight segment BC′, so choose A where BC′ meets l.
For the minimum: A = the point where BC′ cuts l, where C′ is the reflection of C in l
Why it happens: reflecting in l is a mirror move — it preserves every length, so AC and AC′ are equal for every position of A. That turns "make a bent path as short as possible" into "make a path from B to C′ as short as possible", and a straight line is the shortest path between two points.
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