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NCERT Solutions Ganita Prakash (Part 2) Chapter 7 Triangles — In-text Questions

Book page 153 Updated on2026-09-05

Q1.
In the given figure, which triangle has a greater area: ∆XDC or ∆YDC, if both the rectangles are identical?
Answer

Neither — the two triangles have exactly the same area, and each is half of the rectangle.

X and Y both lie on the side AB, and AB ‖ DC. So the perpendicular distance from X to DC and the perpendicular distance from Y to DC are both equal to the width of the rectangle.

Area (∆XDC) = ½ × DC × (distance from AB to DC)
Area (∆YDC) = ½ × DC × (the same distance)
So   Area (∆XDC) = Area (∆YDC) = ½ × area of the rectangle
Why it happens: sliding the apex along a line parallel to the base changes the shape of a triangle but not its height, and the base has not moved either. Area depends only on base and height, so it does not change at all.
Q2.
In the given figure, which triangle has a greater area: ∆XDC or ∆YBC, if both the rectangles are identical?
Answer

They are equal again. Each triangle is half of its rectangle, even though they sit on different sides.

∆XDC: base DC, apex X on AB, height = AD
Area (∆XDC) = ½ × DC × AD = ½ × area of the rectangle

∆YBC: base BC, apex Y on AD, height = AB
Area (∆YBC) = ½ × BC × AB = ½ × area of the rectangle

Since the rectangles are identical, both halves are the same. So Area (∆XDC) = Area (∆YBC).

Why it happens: in a rectangle, opposite sides are parallel and equal. Whichever side you choose as the base, the opposite side is exactly one "height" away — so a triangle with its base on one side and its apex anywhere on the opposite side always fills half the rectangle. The choice of side makes no difference.
Q3.
Find the area of ∆ XDC.
Answer

From Fig. 7.1 the enclosing rectangle ABCD has DC = 5 and AD = 4, and X lies on AB.

base = DC = 5
height = distance from X to DC = AD = 4
Area (∆XDC) = ½ × 5 × 4 = 10 sq. units
Tip: you do not need to know where X sits on AB. Only its distance from DC matters, and that is fixed at 4.
Q4.
To find the area of a triangle, what measurements do we need?
Answer

Just two: one side (the base) and the height to that side — that is, the perpendicular distance from the opposite vertex to the line of that base.

Area of a triangle = ½ × base × height
Why it happens: the triangle is exactly half of the rectangle built on that base with that height (Fig. 7.1). The rectangle needs only its two sidelengths, so the triangle needs only the matching two measurements. The other two sides of the triangle, and its angles, do not enter the calculation at all.
Tip: a triangle has three sides, so it has three base–height pairs. All three give the same area — a fact used again and again in this chapter.
Q5.
How do we get the outer rectangle from the given triangle?
Answer

Draw the line l through the apex A parallel to the base BC. Then drop perpendiculars to l from B and from C.

Step 1: draw l ‖ BC through A
Step 2: at B draw BE ⊥ BC, meeting l at E
Step 3: at C draw CD ⊥ BC, meeting l at D
BCDE is the required rectangle, and ∆ABC lies inside it with the same base BC.
Why it happens: BE and CD are both perpendicular to BC, so they are parallel to each other and equal in length (both equal the distance between the parallel lines l and BC). With ED ‖ BC as well, BCDE has four right angles — it is a rectangle whose base is BC and whose height is the height of the triangle.
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