NCERT Solutions for Class 7th Maths Chapter 6 Constructions and Tilings
Updated on 2026-09-19
About this chapter
Cutting a line segment (or any figure) into two identical parts is called bisection . A line that bisects a segment and is perpendicular to it is the perpendicular bisector . Key property: a point is at the same distance from X and Y exactly when it lies on the perpendicular bisector of XY. Two such points are enough to draw the whole line — that is why the compass construction works. The same idea, applied to a segment whose midpoint is O, gives a 90° angle at O . Bisecting 90° gives 45°; an equilateral triangle gives 60° ; bisecting 60° gives 30° and then 15°. An angle is copied by transferring a chord (SSS congruence) and bisected by cutting equal arcs (SSS congruence). Copying an angle is what lets you construct parallel lines — equal corresponding angles. Angles that add to 360° can b
- Geometric Constructions
- Construction of Perpendicular Bisector
- Construction of a 90° Angle at a Given Point
- Construction Methods in Śulba-Sūtras
- Angle Bisection for a Design
- Steps for Angle Bisection
- Repeating Units and Repeating Angles
- Steps of Construction to Copy an Angle
- Construction of a Line Parallel to the Given Line
- Arch Designs
Quick revision
| Construction | How it is done | What guarantees it | Result |
|---|---|---|---|
| Perpendicular bisector of XY | Equal arcs from X and from Y, above and below; join the two crossing points | ΔABX ≅ ΔABY (SSS), then ΔAOX ≅ ΔAOY (SAS) | OX = OY and ∠AOX = 90° |
| 90° at a point O on a line | Cut X and Y equally on either side of O; one pair of arcs above; join to O | O already lies on the perpendicular bisector of XY | ∠ = 90° |
| 45° angle | Construct 90°, then bisect it | 90° ÷ 2 | 45° |
| Angle bisector of ∠AOB | Mark OA = OB, cut equal arcs from A and B meeting at C, join OC | ΔOBC ≅ ΔOAC (SSS) | ∠BOC = ∠AOC |
| Copy of an angle | Arc from A gives ΔABC; same arc from X; transfer the chord BC to get YZ | ΔABC ≅ ΔXYZ (SSS) | ∠A = ∠X |
| Line parallel to m through B | Copy the angle that transversal l makes with m, at B | Equal corresponding angles ⇒ parallel | m ∥ n |
| 60° angle | Arc from A cuts AX at B; same radius from B cuts the arc at C | ΔABC is equilateral | ∠CAX = 60° |
| 30° and 15° | Bisect 60°, then bisect 30° | Each bisection halves the angle | 30°, 15° |
| Regular hexagon, side s | Six equilateral triangles of side s around one point | 6 × 60° = 360° | Angles all 120° |
| Tiling a grid with 2 × 1 tiles | Cover each column (or row) of even length with tiles | Every tile covers exactly 2 squares | Needs an even number of squares |
| Proving a tiling impossible | Colour black/white like a chessboard and count | Each tile takes one of each colour | Counts must be equal |
| Tiling the plane | Fit copies around every point | 4 × 90° = 6 × 60° = 3 × 120° = 360° | Square, triangle, hexagon |
Exercises
- In-text Questions — Geometric Constructions Page 137
- Eyes of different shapes · Construction of Perpendicular Bisector — In-text Questions Page 139
- Construction of Perpendicular Bisector — Figure it Out Page 140
- Construction of a 90° Angle at a Given Point — In-text Questions Page 141
- Construction Methods in Śulba-Sūtras — Figure it Out Page 142
- Angle Bisection for a Design — In-text Questions Page 143
- Steps for Angle Bisection — In-text Questions Page 144
- Angle Bisection for a Design — Figure it Out Page 144 – 145
- Repeating Units and Repeating Angles — In-text Questions Page 145
- Steps of Construction to Copy an Angle — Figure it Out Page 147
- Construction of a Line Parallel to the Given Line — In-text Questions Page 147
- Construction of a Line Parallel to the Given Line — Figure it Out Page 148
- Arch Designs — In-text Questions Page 149
- A Pointed Arch — In-text Questions Page 150
- A Pointed Arch — Figure it Out Page 151
- Regular Hexagons · Regular Hexagon and Equilateral Triangles — In-text Questions Page 151
- Regular Hexagon and Equilateral Triangles · Construction of a 60° angle — In-text Questions Page 152 – 153
- Related Constructions · 6-Pointed Star — In-text Questions Page 154
- Related Constructions — Figure it Out Page 154 – 155
- Figure it Out — Tiling Page 156
- In-text Questions — Tiling Page 157
- Tiling grids with 2 × 1 tiles — In-text Questions Page 158
- The black-and-white colouring argument — In-text Questions Page 159 – 160
- Are the following tilings possible? — Figure it Out Page 160
- Tiling the Entire Plane — In-text Questions Page 160 – 162