6. Constructions and Tilings

9 topics1 h

Chapter 6 of NCERT Mathematics for Class 7: Constructions and Tilings. Three sections — Bisecting, and building a right angle, Angles you can build without a protractor and Tiling. Nine videos, 1 h in all.

Worked answers to the 25 exercise questions in this chapter →

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Part 1

Bisecting, and building a right angle

9 minThe perpendicular bisector, and the equidistance property that justifies itA compass cannot be told what perpendicular means. All it can do is hold one distance while it turns.9 min · 13 partsहिंदी9 minConstructing a 90° angle at a chosen point on a lineThere is no second construction here. A right angle at a marked point is the perpendicular bisector run backwards.9 min · 13 partsहिंदी9 minA stretched rope as compass and straightedge: the Śulba-Sūtra constructionsA construction is not defined by the instrument you are holding. It is defined by what that instrument refuses to let you do.9 min · 13 partsहिंदी

Part 2

Angles you can build without a protractor

10 minBisecting an angle, and halving 90° to get 45°Bisecting an angle is not a second construction to memorise. It is the first one, wearing different clothes.10 min · 13 partsहिंदी9 minCopying an angle, and why triangle congruence proves it worksA compass carries a length. It cannot carry an angle — there is nothing on the instrument that holds one.9 min · 13 partsहिंदी9 minConstructing 60° from an equilateral triangle, and the arches built on itAn arch is a curve, and a curve is the one thing a hand cannot repeat. So the drawing does not start with the curve.9 min · 13 partsहिंदी9 minRegular hexagons, and why the angles round a point must total 360°The hexagon is never attacked head-on. Instead of asking what its angles are, ask whether six of something can close round a point.9 min · 13 partsहिंदी

Part 3

Tiling

9 minTangrams: rearranging pieces without changing the areaCut one square into seven pieces, allow no gaps and no overlaps, and something odd happens to the outline.9 min · 13 partsहिंदी10 minWhat it takes to cover a region with no gaps and no overlaps“I could not do it” is not an answer. Yes and no are not the same kind of claim, and they cost different amounts to earn.10 min · 13 partsहिंदी

What you will be able to do

  • State what bisection means and what the extra word perpendicular adds
  • State the problem: a right angle wanted at a named point of a given line
  • Name the Śulba-Sūtras as India's earliest surviving texts carrying these construction methods, and say what they were written for
  • Work out, from an eight-armed design, that its supporting lines must sit 45° apart
  • Explain why a compass alone cannot transfer an angle directly
  • Explain why an arch has to be reduced to support lines before it can be drawn
  • State what makes a polygon regular, and name the regular 3-gon and 4-gon
  • Describe how the seven tangram pieces are obtained from one square
  • State what it means for a region to be tiled by a given tile

What this chapter assumes you already know

  • Drawing an arc of a chosen radius from a chosen centre with a compass
  • Triangle congruence, and reading off equal parts once two triangles match (Part II, printed Chapter 1, "Geometric Twins")
  • The Class 6 "Eyes" construction, in which two arc centres were placed by eye
  • The fact that a segment has exactly one perpendicular bisector
  • Setting a compass to a fixed opening and marking equal lengths along a line
  • Knowing that two points determine one line
  • Knowing that a circle is the set of points one fixed distance from a centre
  • Comfort with the idea that the same geometry can be done at different scales

It builds directly on SAS, and why the angle has to be the included one, The four angles at a crossing: vertically opposite and linear pairs, SSS: three sidelengths fix a triangle completely, A transversal creates two matching sets of four angles, Corresponding angles are equal exactly when the lines are parallel and Drawing a parallel line using the corresponding-angle test.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "The arcs above and below must be drawn with the same radius."
  • "You need all four arcs — that is what makes it work."
  • "Both crossings have to be on opposite sides of XY."
  • "It is perpendicular because it looks perpendicular."
  • "Bisecting is just finding the middle by measuring."
  • "Bisection is a thing you do to line segments."
  • "This is a fresh construction to be learnt separately."
  • "You must draw arcs above and below."