NCERT Solutions for Class 7th Maths Chapter 6 Construction Methods in Śulba-Sūtras — Figure it Out
Book page 142 Updated on2026-09-19
Q1.
Justify why AB in Fig. 6.4 is the perpendicular bisector.
The midpoint pulled above XY — mark it A
The midpoint pulled below XY — mark it B
Fig. 6.4, page 142, redrawn — the folded rope tied to pegs at X and Y, its midpoint pulled taut first above XY and then below it.
Answer
Because the rope construction produces exactly the four equal distances we need.
The rope is folded in half, so both halves have the same length, say r.
One loop is fixed at X, the other at Y.
Pull the marked midpoint up to A with both halves fully stretched: AX = r and AY = r ⇒ AX = AY
Pull the same midpoint down to B with both halves stretched: BX = r and BY = r ⇒ BX = BY
So AX = AY = BX = BY
⇒ A and B are both equidistant from X and Y
⇒ both lie on the perpendicular bisector of XY
⇒ AB is the perpendicular bisector of XY
Why it happens: a stretched rope of fixed length is a compass. Fixing one end at X and sweeping the other traces a circle of radius r about X — exactly what the compass leg does on paper. The half-way mark on a taut rope is the same distance from both pegs, which is the one property the whole proof needs.
Did you know? The Śulba-Sūtras are geometric texts of the Vedic period that describe how to lay out fire altars. They belong to the Vedāṅgas — literally the 'limbs of the Vedas'. This rope method is from the Kātyāyana-Śulbasūtra 1.2.
Q2.
Can you think of different methods to construct a 90° angle at a given point on a line using a rope?
Answer
Here are two rope methods. Both need nothing but a rope, pegs and level ground.
Method 1 — make the point a midpoint (the same idea as on paper).
Let O be the given point on the line. Use a short piece of rope as a measure: mark off the same length from O on both sides, giving pegs at X and Y. Now OX = OY.
Take a longer rope, loop its ends at X and Y, fold it to find its midpoint, and pull that midpoint taut to one side. Peg that position as A.
Stretch the rope from A to O. ∠AOX = 90°.
Method 2 — the 3–4–5 rope (an old surveyor's trick).
Knot a rope into a closed loop with 12 equal parts marked on it.
Peg the loop into a triangle whose sides are 3 parts, 4 parts and 5 parts, with the 3-part side and the 4-part side meeting at O along the given line.
The angle at O is a right angle, because 3² + 4² = 9 + 16 = 25 = 5².
Why it happens: Method 1 is the perpendicular-bisector property in rope form — equal distances from two pegs. Method 2 uses the converse of the Pythagoras relation: a triangle whose sides satisfy a² + b² = c² must be right-angled at the corner between a and b. The Śulba-Sūtras contain rope constructions of exactly this kind.
Tip: keep the rope fully taut at every step. A slack rope is a compass with a loose screw — the radius quietly changes and the construction fails.