Geometric Constructions

Class 7 Maths · Chapter 14: Constructions and Tilings · NCERT Ganita Prakash section P2 6.1

Learn

With only a compass and a straightedge we can draw surprisingly many things. Zoya draws an "eye": two arcs through the ends X and Y of a segment, one from a centre above XY and one from a centre below, both the same distance from X and Y. Every such centre lies on one special line.

Perpendicular bisector: open the compass to more than half of XY. Draw arcs from X above and below, then from Y with the same opening, crossing at A and B. Line AB cuts XY in half at 90°. Why? AX = AY and BX = BY, so △AXB ≅ △AYB (SSS). Every point on AB is equally far from X and Y. This also gives a 90° angle.

every point on AB: distance to X = distance to Y

Angle bisector: from the vertex O draw an arc cutting the arms at P and Q. From P and Q, draw equal arcs meeting at R. Then OR splits the angle into two equal halves (△OPR ≅ △OQR by SSS). Halving again and again from 60° and 90° gives 30°, 15°, 45°, 22.5° …; the ancient Śulba-sūtras made such constructions for altars using ropes and pegs.

60° → 30° → 15° · 90° → 45° → 22.5°

Copying an angle: draw an arc from the old vertex cutting both arms, and the same arc from the new vertex. Measure the gap between the arm-crossings with the compass and mark it on the new arc. Joining gives an exact copy, no protractor needed. Copying an angle at a point P on a transversal gives a line through P parallel to the first line (equal corresponding angles).

Hexagons, stars and arches: keep the compass at the circle's radius and step it round the circle: it fits exactly 6 times, giving a regular hexagon whose side equals the radius (six equilateral triangles, so 60° at the centre). A 60° angle comes straight from an equilateral triangle (arc from A, same arc from where it cuts the line), and two of them make 120°, the hexagon's angle. Joining alternate corners gives a 6-pointed star. For a symmetric arch like those in the Red Fort's Diwan-i-Aam, first draw support lines: equal angles at A and D, with AB = CD.

side of hexagon = radius

Explore: bisect an angle.

1. Put the steps for bisecting an angle in order.
2. Now mark a point equally far from A and B.

Remember

• Perpendicular bisector: equal arcs from both ends; its points are equidistant from the ends.
• Angle bisector, copying an angle, parallels and hexagons all use equal arcs and congruent triangles.

Quick check

How many times does the radius step round its own circle?

1. 6
2. 4
3. 8
4. about 3.14

Think about it

How would you draw a perpendicular to line l from a point P not on l?

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.