In-text — Properties of Projections — Exploring Some Geometric Themes

1. 13 marksGanita Prakash Cl-8 Part 2, Math Talk, page 89

   In Fig. 4.3, let l be the actual length of a line and p the length of its projection. Draw AE perpendicular to BC. Compare the lengths p and l. When is the length of the projected line equal to its actual length?

   Hint. AECD is a rectangle, so AE = p. Then look at the triangle AEB.

   Why AECD is a rectangle. AD and EC are both perpendicular to the plane, so they are parallel to each other and equal in length. A quadrilateral with one pair of sides both parallel and equal is a parallelogram, and since the angles at D and C are right angles, it is a rectangle. Therefore AE = DC = p.

   Now use the right triangle. AE was drawn perpendicular to BC, so ∠AEB = 90° and triangle AEB is right-angled at E, with AB = l as its hypotenuse and AE = p as one leg. By the Baudhāyana-Pythagoras theorem, l² = p² + EB²

   Since EB² is never negative, l² ≥ p², and therefore l ≥ p.

   So the projection is never longer than the line itself. Projecting can shorten a segment but never stretch it — which matches the everyday observation that a stick's shadow is at most as long as the stick when the light comes straight on.

   When are they equal? Equality needs EB = 0, that is, the point E coincides with B. That happens exactly when AB is parallel to the plane, so the segment has no component running towards or away from it. In that case the segment and its projection are the two long sides of a rectangle, and are equal.

   And the other extreme. If AB is perpendicular to the plane, the whole segment projects to a single point and p = 0 — the greatest possible shortening.

   ✦ AECD is a rectangle so AE = p, and in the right triangle AEB the hypotenuse is l, giving l² = p² + EB² and hence p ≤ l always. They are equal exactly when the line is parallel to the plane, and the projection shrinks to a point when the line is perpendicular to it.

2. 24 marksGanita Prakash Cl-8 Part 2, Math Talk, pages 89 and 94

   What are the possible projections of a square under different orientations? Of a parallelogram? Can the projection of a parallelogram ever be a quadrilateral that is not a parallelogram? What about a regular n-sided polygon? [Hint: start with a pair of parallel lines.]

   Hint. Work out what happens to a pair of parallel lines first — everything else follows from that.

   Start with the hint — parallel lines. Take two parallel segments. Project them onto a plane. The projecting lines from each segment sweep out two parallel planes, and two parallel planes cut the projection plane in two parallel lines. So:

   Parallel lines always project to parallel lines.

   Moreover, projection scales all lengths in a given direction by the same factor, so two equal parallel segments project to two equal parallel segments. You can verify this outdoors: hold a parallelogram cut from card in sunlight, and however you turn it, its shadow stays a parallelogram.

   Projection of a parallelogram. Its opposite sides are parallel and equal, so their projections are parallel and equal too. A quadrilateral whose opposite sides are parallel and equal is a parallelogram. Therefore:

   The projection of a parallelogram is always a parallelogram — a rectangle, rhombus or square in special positions, and squashed to a line segment when the parallelogram is perpendicular to the plane. It can never be a trapezium or any other non-parallelogram quadrilateral.

   Projection of a square. A square is a special parallelogram, so its projection is a parallelogram: a square when the square is parallel to the plane, a rectangle when it is turned about one of its sides, a general parallelogram in a slanted position, and a line segment when it stands perpendicular to the plane. What it can never be is a quadrilateral with non-parallel sides.

   Projection of a regular n-gon. Its projection is composed of the projections of its n sides, so in general it is an n-sided polygon, but usually not regular — the sides get shortened by different amounts depending on their direction. What survives is: · any pair of parallel sides stays parallel (so for even n, opposite sides remain parallel and equal); · the polygon stays convex; · if the polygon's plane is parallel to the projection plane, the projection is a congruent copy; · if its plane is perpendicular, it collapses to a line segment.

   The idea underneath. Projection destroys lengths and angles but preserves parallelism and straightness. Anything defined only by parallelism — being a parallelogram, having parallel opposite sides — survives. Anything defined by equal lengths or equal angles — being regular, being a square — does not.

   ✦ Parallel lines project to parallel lines, so a parallelogram always projects to a parallelogram (or to a line segment) and never to a non-parallelogram quadrilateral; a square projects to a square, rectangle, general parallelogram or line segment; and a regular n-gon projects to an n-sided convex polygon that is usually not regular, congruent only when its plane is parallel to the projection plane.

3. 33 marksGanita Prakash Cl-8 Part 2, Math Talk, pages 90-91 and 94

   Why do we take three mutually perpendicular projections of an object rather than one? And what is the connection between a projection and a shadow?

   Hint. Ask whether you could rebuild the object from a single view.

   One projection is not enough, because it does not determine the object. The chapter shows several different lines that give the same projection, and several different cuboids that also share a projection. A circular projection could come from a sphere, a cylinder seen end-on, a cone seen from above, or a flat disc. Projection throws information away — specifically, everything about the direction being flattened.

   So a single view can never be relied on to identify a solid.

   The fix — three mutually perpendicular views. Take a plane in front of the object (the vertical plane), one below it (the horizontal plane) and one to its side (the side plane). The projections onto these are called the

   · front view — on the vertical plane · top view — on the horizontal plane · side view — on the side plane

   Each view records two of the three dimensions and loses the third, but a different third each time. Between them, every direction is captured by at least two views, and the three together usually pin the object down. This is the formal version of the "profiles" explored earlier in the chapter, and it is why engineering drawings always carry three views.

   The connection with shadows. Shine a torch at an object in front of a wall and the shadow looks much like the projection — but not exactly. It is scaled up, and slightly stretched or distorted, because the torch's rays spread out from a point. Move the torch further back and the shadow shrinks towards the true projection, because distant rays arrive more nearly parallel.

   Now imagine a torch infinitely far away. Its rays would be exactly parallel, and the shadow would be indistinguishable from the projection. Such a torch exists: the Sun. When sunlight falls perpendicular to a plane, the shadows it casts on that plane are projections.

   That makes the whole theory testable outdoors with a cardboard cutout — which is how the chapter suggests confirming that a parallelogram's shadow is always a parallelogram.

   ✦ Because a single projection does not determine the object — many different solids share one view — three mutually perpendicular projections are used: the front view on the vertical plane, the top view on the horizontal plane and the side view on the side plane. A shadow is a projection distorted by the light spreading from a point, so a distant source such as the Sun casts shadows that are effectively exact projections.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 8 Mathematics textbook Ganita Prakash Part 2, Reprint 2026-27 (hegp204.pdf), where this is Chapter 4 (pages 70-102) — the eleventh chapter of the Class 8 course and the longest in the book. Like the rest of Part 2 it carries NO printed answer key, so every formula and count was derived and then independently recomputed in Python before being written: the Sierpinski Carpet recurrences (R_n = 8^n, H_n = (8^n - 1)/7), the Sierpinski Triangle counts (3^n and (3^n - 1)/2), the areas (8/9)^n and (3/4)^n, the Koch side count 3 x 4^n and perimeter 3 x (4/3)^n, and the face/edge/vertex formulas for prisms and pyramids (checked against Euler's relation for every case). THIS IS A HEAVILY VISUAL CHAPTER, so figure-only items are handled in one of two ways and never guessed. (1) MEASURED FROM THE RENDERED PAGE: the cube-stack count on page 97 was settled by rendering the figure at 400 dpi and observing that each row sits one step BACK as well as one step up (every bottom cube shows its full top face), which makes it a square-layered step pyramid of 16 + 9 + 4 + 1 = 30 cubes rather than the ten visible; and the three letters in the page-96 puzzle were read off the printed pixel glyphs at 700-900 dpi as C (front), A (top) and F (side). (2) FLAGGED AND ANSWERED BY METHOD: the six candidate cube nets, the projection-matching sets, the cube-combination views, the isometric figures to copy, the rolling ball and the impossible triangle are all printed diagrams; each solution gives the full method and reasoning and says plainly that the diagram is not reproduced. ONE ITEM IS LEFT OPEN BY THE BOOK ITSELF and is reported as such: the 30 x 12 x 12 shortest-path Try This on page 87, where the book computes 42 cm and 40 cm for two unfoldings (24^2 + 32^2 = 1600 verified) and then says all unfoldings must be listed to find the answer — so the solution establishes only that the shortest path is at most 40 cm. The tetracube count in the page-100 exercise was verified by exhaustive computer enumeration: 8 arrangements up to rotation, 7 up to rotation and reflection, of which 5 are flat.. Questions are referenced from the NCERT textbook for identification.