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NCERT Solutions Ganita Prakash (Part 2) Chapter 7 Some Applications of the Area Formula — In-text Questions

Book page 155 Updated on2026-09-05

Q1.
Find BY.
Answer

In the figure, AX ⊥ BC with AX = 5, BC = 3, and AC = 4, while BY ⊥ AC.

The trick is to compute the same area twice, once with each base–height pair.

Using base BC: Area (∆ABC) = ½ × AX × BC = ½ × 5 × 3 = 15/2 sq. units

Using base AC: Area (∆ABC) = ½ × BY × AC = ½ × 4 × BY = 2 BY

So   2 BY = 15/2
BY = 15/4 = 3.75 units
Why it happens: a triangle has one area but three base–height pairs, and all three must give the same number. Writing that equality down turns a known area into an unknown altitude. This "two ways, one area" move is used again in Q2 and Q3 of the next exercise.
Q2.
What is Area (∆ABC)?
Answer

Use the base and height that are already marked in the figure.

Area (∆ABC) = ½ × AX × BC
= ½ × 5 × 3
= 15/2 = 7.5 sq. units
Tip: AX is the perpendicular from A to BC, so BC and AX are a matching base–height pair. Never multiply a base by a slanted side.
Q3.
Are the 4 triangles obtained by drawing the diagonals of a rectangle (regions 1 – 4 in the figure) of equal areas?
Answer

Yes — all four have equal areas, even though they are not all congruent.

Let the diagonals meet at O. Take two adjacent triangles, say 1 and 2, and choose OD and OB as their bases. Both bases lie on the same straight line BD, and both triangles have the same apex A, so they have the same height — the perpendicular from A to BD.

The diagonals of a rectangle bisect each other, so OB = OD
Same base length, same height → Area(1) = Area(2)
Repeating around the point O: Area(2) = Area(3), Area(3) = Area(4)
Area(1) = Area(2) = Area(3) = Area(4) = ¼ × area of the rectangle
Why it happens: triangles 1 and 3 are congruent to each other, and so are 2 and 4 — but a triangle from the first pair need not be congruent to one from the second. Equal area is a weaker demand than congruence, and the bisected diagonals supply exactly that. This gives the general statement: in a triangle, the line joining a vertex to the midpoint of the opposite side divides it into two triangles of equal areas.
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