Class 7 | Maths
Chapter 6 | Constructions and Tilings
Aligned with CBSE | NCERT | School & Olympiad Standards
Focus on What Matters Most
The67Sprint
π Perpendicular Bisector
- Bisection means dividing a line segment or angle into two exactly identical parts
- Draw equal-radius arcs above and below the segment from both endpoints X and Y β connecting the intersection points gives the perpendicular bisector
- Radii for the arcs above and below can differ, as long as each pair from X and Y uses the same radius
- Any point on this bisector is equidistant from both endpoints
- The compass radius must stay locked between the two arcs of a pair β nudging it breaks the construction
π Building Angles Without a Protractor
- An equilateral triangle drawn with a compass automatically gives a perfect 60Β° angle
- Bisecting angles with intersecting arcs turns 90Β° into 45Β°, and 60Β° into 30Β° and 15Β°
- 60Β° on a straight line leaves a supplementary 120Β° angle
- Angles can be exactly copied using SSS congruence β measure the chord between the two arc intersections on the arms, not just "distance between arms"
- Parallel lines are drawn by copying the corresponding angle across a transversal
⬑ Regular Hexagons
- A regular hexagon is made of six congruent equilateral triangles meeting at a central point
- Six 60Β° angles sum to exactly 360Β°, so the triangles close up with zero gaps
- Each interior angle of a regular hexagon measures 120Β°
𧩠Tiling the Basics
- Tiling means covering a flat region with zero gaps and zero overlaps
- Each 2Γ1 tile always covers exactly 2 squares β an even number
- So any grid with an odd total number of squares (like 5Γ7 = 35) can never be tiled with 2Γ1 tiles
- A Tangram is an ancient Chinese puzzle of 7 pieces cut from a single square, used to rearrange shapes into new figures
Tangram π§©
Golden Rule of the Compass
Once a compass is opened for the first arc of a bisector or angle copy, never change its width for the second arc. Equal radii are what make the construction mathematically exact.
Quick Recall β Sprint Facts
120Β° Angle
Supplementary to a constructed 60Β° angle on a straight line
30Β° & 15Β° Angles
Made by successively bisecting a 60Β° angle
Hexagon Interior Angle
120Β° β the sum of two adjacent 60Β° triangle corners
Odd Total = No Tiling
2Γ1 tiles cover 2 squares each β an odd grid total can never be tiled
Εulba-SΕ«tras
Ancient Indian texts using stretched ropes to find exact perpendiculars
Tangram
7-piece ancient Chinese puzzle cut from a single square
π Key Terms
Bisection
Dividing a line, angle, or any geometrical object into two exactly identical parts
Perpendicular Bisector
A line cutting a segment exactly in half at a 90Β° angle β every point on it is equidistant from the endpoints
Tiling (Tessellation)
Covering a flat region using shapes with zero gaps and zero overlaps
Congruence
Two figures that are exactly the same size and shape β the basis for angle copying (SSS)
Εulba-SΕ«tras
Ancient Indian mathematical texts describing rope-based methods for exact geometric constructions
Tangram
A 7-piece ancient Chinese puzzle formed by dividing a square, used to explore shape rearrangement
π Olympiad & Challenger Edge
Algebraic m Γ n Parity
For any m Γ n grid: if both m and n are even, it tiles exactly with (m/2) Γ n vertical 2Γ1 tiles. If both are odd, the total area is odd, breaking the rule that every tile covers 2 squares.
Copying an Unmeasurable Angle
You cannot construct a 65.5Β° angle from scratch with ruler and compass, but you CAN copy an existing 65.5Β° angle exactly β angle copying relies on SSS congruence, independent of the actual degree measure.
Hidden Triangle in a Hexagon
Joining alternate vertices of a regular hexagon (AβCβE) forms an equilateral triangle: 180Β° β 120Β° = 60Β° remains at each new vertex, giving three equal 60Β° angles.
M.C. Escher's Tessellations
The Dutch artist proved that non-regular, complex interlocking shapes β fish, birds, lizards β can tile a plane perfectly, not just squares, triangles, and hexagons.
β Radius Changed Mid-Arc?
Wrong. The radius must stay strictly locked for both arcs in a pair from X and Y, or the intersection won't be equidistant.
β Need a Protractor for 45Β°?
No. Draw a 90Β° angle and bisect it with intersecting arcs β examiners want to see those construction marks.
β Even Squares Always Tile?
Not necessarily. A grid can have an even total yet still fail the checkerboard color-matching test.
β Erasing the Faint Arcs?
Never. Construction arcs are the proof you used geometry β not a protractor or scale β and marks are deducted if they're missing.
ποΈ Diwan-i-Aam, Red Fort
The Trefoil and Pointed Arches here were built using equal support lines and intersecting arcs β the exact method taught in this chapter.
π Honeycomb Engineering
Bees build hexagonal cells because regular hexagons tile perfectly (360Β° at each vertex), wasting zero wax and maximising honey storage.
π Checkerboard Coloring
Coloring a grid black-and-white shows every 2Γ1 tile must cover one of each color β a foundational parity trick used in computer science and puzzle design.
ποΈ The Kanizsa Triangle
Cutting 60Β° wedges from circles and aligning them tricks the brain into seeing a solid white triangle that was never actually drawn.
π¨ Isthmus of Geometry Class
Missing intersection labels (like point O) or the 90Β° symbol on a bisector costs marks β always label your construction.
π¨ Trefoil vs. Pointed Arch
Both need support lines of equal length AND equal corresponding base angles β symmetry doesn't happen by eye alone.
π¨ "14 Squares Must Tile"
A 5Γ3 grid with one square removed has 14 squares (even) β but if the removed square breaks the color balance, it still cannot be tiled.
π¨ Angle Copying β Measuring
Copying an angle means transferring the chord length with a compass β not eyeballing or estimating the degree value.
Logic Chain β How a Perpendicular Bisector Is Constructed
Step 1
Identify the segment's endpoints X and Y β this sets up the two reference points every arc will measure from
Step 2
Lock the compass past half of XY, then draw matching arcs above and below from both X and Y, which leads to two crossing points
Step 3
Those crossing points are equidistant from X and Y, which connects directly to the line joining them
Step 4
Drawing that line produces the perfect 90Β° perpendicular bisector, proven by the equal arcs rather than any measurement
π¦
Mentor Byte
"A compass is not a circle-drawing tool β it is a distance-locking machine. Every arc you swing is a silent proof of equal distance, and every crossing point is the math keeping its word. Numbers on a ruler tell you what to trust blindly; an intersecting arc shows you why. Leave the faint marks on the page. They are not clutter β they are the argument itself."