1. Chapter at a Glance
With only a ruler and compass, you can bisect a line, bisect an angle and draw angles of 60°, 90°, 30° and 45° — no protractor needed. With repeated shapes, you can cover a floor completely without gaps or overlaps, like the tiles in your home or the patterns in rangoli and temple floors. This chapter combines both ideas: precise constructions and tilings (tessellations).
- Constructing the perpendicular bisector of a segment.
- Constructing the bisector of an angle.
- Constructing 60°, 120°, 90°, 30° and 45° angles with a compass.
- Drawing a regular hexagon inside a circle.
- Tilings: which shapes tile a plane and why angles at a point must add to 360°.
- Regular and semi-regular tilings in art and architecture.
2. Perpendicular Bisector of a Segment
The perpendicular bisector of AB is the line that passes through the midpoint of AB at 90°. Every point on it is equally far from A and B.
- Draw AB. Open the compass to more than half of AB.
- With centre A, draw arcs above and below AB.
- With centre B and the same radius, draw arcs cutting the first arcs at P and Q.
- Join PQ. It meets AB at M, the midpoint, and PQ ⊥ AB.
3. Bisecting an Angle
To bisect ∠ABC (divide it into two equal angles):
- With centre B and any radius, draw an arc cutting BA at P and BC at Q.
- With centres P and Q and the same radius (more than half of PQ), draw arcs meeting at R inside the angle.
- Join BR. It is the angle bisector: ∠ABR = ∠RBC.
Why it works: ΔBPR and ΔBQR are congruent by SSS (BP = BQ, PR = QR, BR common), so the two angles at B are equal.
4. Constructing Special Angles
| Angle | Method |
|---|---|
| 60° | Draw an arc from O cutting the ray at P; from P with the same radius cut the arc at Q. ∠POQ = 60° (ΔOPQ is equilateral). |
| 120° | Step the same radius once more along the arc from Q to R. ∠POR = 120°. |
| 30° | Bisect 60°. |
| 90° | Bisect the angle between the 60° and 120° marks, or construct the perpendicular at a point. |
| 45° | Bisect 90°. |
| 15°, 75°, 135° | Bisect 30°; 60° + 15°; 90° + 45°. |
Regular hexagon in a circle
Draw a circle. Keep the compass at the same radius and step around the circle starting from any point — exactly six steps return to the start. Joining the six points gives a regular hexagon whose side equals the radius (it is made of six equilateral triangles).
5. Tilings (Tessellations)
A tiling covers a flat surface with shapes that fit together with no gaps and no overlaps. Where tiles meet at a point, the angles around that point must add up to exactly 360°.
Which regular polygons tile alone?
| Regular polygon | Each interior angle | 360° ÷ angle | Tiles alone? |
|---|---|---|---|
| Equilateral triangle | 60° | 6 | Yes |
| Square | 90° | 4 | Yes |
| Regular pentagon | 108° | 3.33… | No |
| Regular hexagon | 120° | 3 | Yes |
| Regular octagon | 135° | 2.67… | No |
Only three regular polygons — the triangle, square and hexagon — tile a plane by themselves. Bees use hexagons in honeycombs because hexagons tile without gaps and use the least wax for a given area.
Semi-regular tilings and art
Different regular polygons can be combined: two octagons and a square fit at a point (135° + 135° + 90° = 360°), a pattern often seen on floors. Triangles and hexagons, or squares and triangles, also combine. Indian architecture — from step-wells to temple floors and jali screens — and rangoli patterns use such tilings beautifully.
Any triangle and any quadrilateral (even irregular ones) can tile the plane, because their angles can be arranged to make 360° at every point (a quadrilateral's angles add to 360°; two copies of a triangle's angles make 360°).
6. Solved Problems (NCERT Style)
How can you construct a 75° angle with ruler and compass only?
Solution: Construct 60° and 90° at the same point on the same ray, then bisect the angle between them (30°) to get 15°. 60° + 15° = 75°.
Can one regular hexagon and two squares meet at a point in a tiling?
Solution: 120° + 90° + 90° = 300° ≠ 360°. A gap of 60° remains, which an equilateral triangle could fill (hexagon + 2 squares + triangle = 360°). Alone, the three shapes do not close the gap.
Find the midpoint of a 7.3 cm segment without measuring.
Solution: Construct its perpendicular bisector; where it cuts the segment is the midpoint (3.65 cm from each end).
7. Master Formula Table
| Fact | Rule |
|---|---|
| Perpendicular bisector | points equidistant from A and B |
| Angle bisector | divides angle into two equal parts |
| 60° with compass | equilateral triangle from equal radii |
| 90°, 45°, 30°, 15° | bisect 180°, 90°, 60°, 30° |
| Hexagon side in a circle | = radius |
| Tiling condition | angles at a point = 360° |
| Interior angle of regular n-gon | 180° × (n − 2) ÷ n |
| Regular tilings | only n = 3, 4, 6 |
8. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Perpendicular bisector | Equal arcs from both ends | Midpoint, equidistance |
| Angle bisector | Arcs from points on arms | Bisect given angles |
| Special angles | 60° base, then bisect | 30°, 45°, 75°, 135° |
| Hexagon | Six radii around a circle | Construction steps |
| Tilings | No gaps, no overlaps; 360° | Which shapes tile |
9. Mind Map
- Of a segment
- Of an angle
- 60°, 120°
- 90°, 45°, 30°
- Hexagon in circle
- Interior angles
- 360° at a point
- Triangles, squares, hexagons
- Semi-regular patterns
10. Tips and Tricks
- Keep construction arcs visible — examiners award marks for method.
- Use a radius more than half the segment, or the arcs won't meet.
- To test a tiling, add the angles meeting at one point — they must total 360°.
- Changing the compass radius between the two arcs.
- Thinking regular pentagons can tile a floor alone.
- Bisecting 60° and calling it 45°.
11. Important One-liners
- The perpendicular bisector of a segment passes through its midpoint at 90°.
- An angle bisector splits an angle into two equal angles.
- Equal compass steps from a point on a circle make 60° at the centre.
- The side of a regular hexagon in a circle equals the radius.
- Angles meeting at a point in a tiling add up to 360°.
- Only equilateral triangles, squares and regular hexagons tile by themselves.
- Every triangle and every quadrilateral can tile the plane.
12. Exam-pattern Questions with Detailed Answers
Which regular polygon CANNOT tile a floor by itself?
- Square
- Equilateral triangle
- Regular pentagon
- Regular hexagon
Show answer
Its angle 108° does not divide 360° exactly (360 ÷ 108 is not whole).
Answer: (C)
Which angle do you get by bisecting a 90° angle?
Show answer
Half of 90° is 45°.
Answer: 45°
How many equilateral triangles meet at a point in a triangular tiling?
Show answer
360° ÷ 60° = 6.
Answer: 6
Write the steps to construct a 135° angle.
Show answer
On a straight line through O, construct the perpendicular (90°). Bisect the 90° angle on the other side, between the perpendicular and the ray going the other way, to get 45°. 90° + 45° = 135°.
Answer: 90° + bisected 90° (45°) = 135°
Find the interior angle of a regular octagon and check whether two octagons and a square can meet at a point.
Show answer
Interior angle = 180° × 6 ÷ 8 = 135°. 135° + 135° + 90° = 360° ✔.
Answer: 135°; yes, they fit
A circle has radius 4 cm. What is the perimeter of the regular hexagon constructed inside it with compass steps?
- 16 cm
- 24 cm
- 32 cm
- 12 cm
Show answer
Side = radius = 4 cm; perimeter = 6 × 4 = 24 cm.
Answer: (B) 24 cm
Why is every point on the perpendicular bisector of AB equidistant from A and B?
Show answer
For a point P on it, with M the midpoint: ΔPMA ≅ ΔPMB by SAS (AM = BM, ∠PMA = ∠PMB = 90°, PM common). So PA = PB.
Answer: By SAS congruence
How many squares and triangles meet at a point if 2 squares are used and the rest are equilateral triangles?
Show answer
2 × 90° = 180°; remaining 180° ÷ 60° = 3 triangles. So 2 squares and 3 triangles.
Answer: 3 triangles
Show that a regular 12-gon (angle 150°) and an equilateral triangle can meet at a point. How many of each?
Show answer
Two 12-gons: 300°; one triangle: 60°; total 360° ✔.
Answer: Two 12-gons and one triangle
Explain why any quadrilateral can tile the plane.
Show answer
The four angles of any quadrilateral add to 360°. By rotating copies through half-turns about the midpoints of the sides, the four different angles can be placed around each vertex, filling exactly 360° with no gap.
Answer: Its angles sum to 360°, which fills a point exactly
Using only compass steps, how would you construct a 150° angle?
Show answer
Construct 120° (two 60° steps), then construct 30° beyond it by bisecting the next 60° step (from 120° to 180°). 120° + 30° = 150°.
Answer: 120° + 30° = 150°
Why do honeycombs use hexagons rather than triangles or squares, if all three tile?
Show answer
Among shapes that tile, the regular hexagon encloses the largest area for a given perimeter, so the bees use the least wax for storage space. It is also strong and closest to a circle.
Answer: Maximum space for the least wall material
13. Quick Revision Summary
- Perpendicular bisector: equal arcs from both ends; gives midpoint and equidistant points.
- Angle bisector: arcs on the arms, then equal arcs inside; proved by SSS.
- 60° from equal compass steps; bisect for 30°, 90°, 45°, 15°.
- Hexagon in a circle: six steps of the radius.
- Tiling: angles at a point = 360°; regular triangles, squares, hexagons tile alone.