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NCERT Solutions Ganita Prakash (Part 2) Chapter 4 –91Representation of Solids on a Plane Surface — In-text Questions

Book page 89 Updated on2026-09-05

Q1.
What happens to the length of a line in its projection?
Answer

It never gets longer. The projection is at most as long as the segment itself, and usually shorter.

In Fig. 4.3 the segment AB has length l and its projection DC has length p. Drawing AE perpendicular to BC makes AECD a rectangle (AD and EC are both perpendicular to the plane, so AD ∥ EC and AD = EC; that forces AE ∥ DC and AE = DC). So AE = p, and ∠AEB = 90°.

In right triangle AEB, AB is the hypotenuse
AB = l, AE = p
l² = p² + BE² ⇒ l² ≥ p²
p ≤ l
Why the hypotenuse is the longest side: BE² is a square, so it is never negative. It is zero only when B and E coincide — that is, only when the segment already lies parallel to the plane. Every tilt out of the plane adds something to BE² and therefore makes l strictly bigger than p.
Q2.
Can you now compare the lengths p and l?
Answer

p ≤ l, always.

l² = p² + BE²
so l² − p² = BE² ≥ 0
hence p ≤ l

Concretely: if a 10 cm pencil is tilted at 60° to the wall it is being projected on, its shadow measures 10 × cos 60° = 5 cm. Tilt it more and the shadow gets shorter still; hold it flat against the wall and the shadow is the full 10 cm.

Tip: In general p = l cos θ, where θ is the angle the segment makes with the plane. Since cos θ is never more than 1, p can never beat l.
Q3.
When is the length of the projected line equal to its actual length?
Answer

Exactly when the segment is parallel to the plane (or already lying in it).

p = l ⇔ BE = 0 ⇔ B coincides with E
⇔ AB ⊥ AD, i.e. AB is perpendicular to the projecting direction
⇔ AB is parallel to the plane

In that case ABCD is itself a rectangle, so DC = AB exactly.

Why this is the one and only case: tilting the segment out of the plane means one end is further from the plane than the other. Their projections then move closer together than the ends themselves are, and length is lost. Only when both ends are at the same distance from the plane is nothing lost.
Q4.
What do you think are the different possible projections of a square that we get based on its orientation?
Answer

The projection of a square can be a square, a rectangle, a parallelogram, or a line segment — but never anything else.

Orientation of the squareProjection
Parallel to the planeA square of the same size
Tilted about one of its sidesA rectangle — one pair of sides keeps its length, the other pair shortens
Tilted about a diagonal, or generallyA parallelogram (a rhombus in the diagonal case)
Perpendicular to the planeA line segment — the square collapses
Why it is always a parallelogram: the square has two pairs of parallel, equal sides. Projection carries parallel segments to parallel segments and equal parallel segments to equal parallel segments. So the two pairs stay parallel and stay equal in the picture — and a quadrilateral with both pairs of opposite sides parallel is a parallelogram.
Check it yourself: cut a square from card and hold it in sunlight over a sheet of paper. Turn it slowly. You will see the square stretch into rectangles and lean into parallelograms, and just before it disappears the shadow is a thin line — but the opposite sides stay parallel throughout.
Q5.
What do you think is the projection of a parallelogram under different orientations? Can this ever be a quadrilateral that is not a parallelogram?
Answer

The projection of a parallelogram is always another parallelogram (or, in the flattest case, a line segment). It can never be a quadrilateral that is not a parallelogram.

Start where the hint suggests, with a pair of parallel lines. Two parallel lines and the direction of projection lie in two parallel planes. Those planes meet the projection plane in two parallel lines. So parallel lines project to parallel lines.

AB ∥ DC ⇒ their projections A′B′ ∥ D′C′
AD ∥ BC ⇒ their projections A′D′ ∥ B′C′
Both pairs of opposite sides parallel ⇒ A′B′C′D′ is a parallelogram

So a trapezium, a kite or an ordinary quadrilateral can never be the shadow of a parallelogram.

Try This: Cut out a parallelogram and look at its shadow in sunlight, which is as good as a projection because the Sun’s rays are effectively parallel. However you turn it, the shadow stays a parallelogram — it may become a rectangle, a square, a rhombus or a very thin sliver, but the opposite sides never stop being parallel.
Q6.
What can you say about the projection of an n-sided regular polygon? [Hint: Projection of a polygon is composed of the projections of its sides.]
Answer

It is again a convex polygon with n sides — but it is normally not regular. It is regular only when the polygon is parallel to the plane.

Following the hint: the projection of the polygon is built from the projections of its n sides. Each side projects to a segment, corners project to corners, and the pieces stay joined in the same order. So the picture is an n-sided polygon (unless the polygon is perpendicular to the plane, when it flattens to a segment).

What survives and what does not:

PropertySurvives projection?
Number of sidesYes — still n
Sides being parallelYes
ConvexityYes
Midpoints staying midpointsYes
All sides equalNo — sides across the tilt shrink more
All angles equalNo

For an even n the regular polygon has n⁄2 pairs of opposite sides that are parallel and equal, and both facts survive. So the projection of a regular hexagon is a hexagon whose opposite sides are still parallel and equal, even though it looks squashed.

Why regularity is lost: tilting shortens lengths by different factors in different directions — a side pointing along the tilt shrinks most, a side across the tilt not at all. Equal sides that point in different directions therefore end up unequal.
Q7.
How would the projections of a cube and a cone look?
Answer

Cube (Fig. 4.4). Held with one face parallel to the plane, its projection is a square — the near face and the far face land exactly on top of each other, and the four side faces project onto the edges.

Cone (Fig. 4.5). Held with its axis parallel to the plane, its projection is a triangle — the base circle flattens to a segment (the base of the triangle) and the two extreme slant lines give the other two sides.

SolidOrientationProjection
CubeA face parallel to the planeSquare
CubeTilted about one edge directionRectangle
CubeBalanced on a cornerRegular hexagon
ConeAxis parallel to the planeIsosceles triangle
ConeAxis perpendicular to the planeCircle
Why the outline is all we get: the projection records where the solid blocks the projecting rays. Two points of the solid on the same ray land on the same spot, so everything behind the outline is lost. That is why a solid and a hollow shell of the same shape have identical projections.
Q8.
See Figures 4.2 – 4.5. In each case, see if you can visualise another object that gives the same projection.
Answer

Yes — in every case there are many others. Fig. 4.6 shows two families of examples.

FigureObject shownAnother object with the same projection
4.2A segment tilted to the planeAny longer segment tilted more steeply — a 5 cm segment at 53° and a 4 cm segment at 41° both project to 3 cm
4.3A segment with projection pThe same segment slid anywhere parallel to the plane, or turned about the projecting direction
4.4A cube, projecting to a squareA cuboid of any depth with the same square face; a square pyramid seen from above; a hollow box
4.5A cone, projecting to a triangleA triangular prism seen end on; a square pyramid seen from the side; a flat triangular cutout
Cuboids 1, 2 and 3 units deep — all give the same square front view
Lines of different lengths and tilts — all give the same segment
Why this is worth noticing: it is the reason engineers never trust one drawing. A single projection throws away everything along the direction of projection. Take three projections on mutually perpendicular planes and far less is lost — though, as Question 9 shows, even three views do not always pin the object down.
Q9.
Find another object that makes the same projection as that of a given cone.
Answer

Held with its axis parallel to the plane, a cone projects to an isosceles triangle. So does each of these:

  • A triangular prism whose end face is that triangle, seen along its length.
  • A square pyramid of the same height and base width, seen from the side.
  • A flat triangular sheet of card of exactly that shape, held parallel to the plane.
  • A hollow cone, or a cone of any material at all — the projection cannot tell.
Why so many: the projection is only the outline of the shadow. Any solid that fits neatly inside the ‘tube’ of rays which the cone fills, and blocks all of it, casts the same shadow. Depth, thickness and what is inside are all invisible. That is precisely the loss of information the three standard views are designed to reduce.
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