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Constructions and Tilings – Class 7 Mathematics notes, important questions and MCQs

NCERT Class 7 Mathematics, chapter 14. Notes, key terms, important questions with answers and MCQs.

Chapter overview

With a ruler and compass we can construct perpendicular bisectors, angle bisectors and perpendiculars without measuring angles. Tiling means covering a flat surface with shapes without gaps or overlaps; only some regular polygons can tile a plane on their own.

Geometric constructions

Perpendicular bisector of AB: with radius more than half AB, draw arcs from A and B on both sides; join the two crossing points.
Angle bisector: from the vertex draw an arc cutting both arms; from these two points draw equal arcs that cross; join the vertex to the crossing point.
A 60° angle is constructed by drawing an arc from the vertex and cutting it with an arc of the same radius from the point where it meets the arm.

Tilings

A tiling (tessellation) covers a surface completely with shapes, without gaps or overlaps.
At every meeting point of tiles, the angles must add up to 360°.
Among regular polygons, only equilateral triangles (6 × 60°), squares (4 × 90°) and regular hexagons (3 × 120°) tile a plane by themselves.
Regular pentagons (108°) cannot tile alone because 360 is not a multiple of 108; combinations such as octagons with squares (135° + 135° + 90°) can tile.

Key terms

Perpendicular bisector: a line that passes through the midpoint of a segment at 90° to it
Angle bisector: a ray that divides an angle into two equal angles
Tiling: covering a flat surface with shapes without gaps or overlaps
Regular polygon: a polygon with all sides equal and all angles equal

Formulas

Interior angle of a regular n-gon = (n − 2) × 180° ÷ n
At a tiling vertex: sum of angles = 360°

Important questions with answers

What is a tiling? Name the regular polygons that can tile a plane by themselves.2 marks

A tiling is a covering of a flat surface by shapes without gaps or overlaps. The only regular polygons that can tile a plane by themselves are the equilateral triangle, the square and the regular hexagon.

Write the steps to construct the bisector of an angle ABC.3 marks

With B as centre and any radius, draw an arc cutting BA at P and BC at Q. With P and Q as centres and a radius more than half of PQ, draw two arcs that cross at R inside the angle. Join BR. The ray BR is the bisector of angle ABC, so angle ABR equals angle RBC.

Explain why only three regular polygons can tile a plane, using their interior angles.5 marks

For tiles to fit around a point without gaps or overlaps, the angles meeting there must add up to exactly 360°. If only one kind of regular polygon is used, 360° must be a whole-number multiple of its interior angle. The equilateral triangle has 60°, and 6 × 60° = 360°. The square has 90°, and 4 × 90° = 360°. The regular hexagon has 120°, and 3 × 120° = 360°. The regular pentagon has 108°; 3 × 108° = 324° leaves a gap and 4 × 108° = 432° overlaps. Polygons with more than six sides have interior angles greater than 120° but less than 180°, so fewer than three would be needed, which is impossible since at least three tiles must meet at a point. Hence only these three regular polygons tile a plane by themselves.

Quick revision

Perpendicular bisector: equal arcs from both ends.
Angle bisector: arc on arms, then equal arcs, join vertex.
Regular tilings: triangle, square, hexagon only.
Angles around a point = 360°.

Exam tips

In constructions, keep all arc marks; examiners check the construction lines.
To test if a regular polygon tiles alone, check whether 360 ÷ (interior angle) is a whole number.

Practice MCQs

Covering a floor with shapes without gaps or overlaps is called
1. tiling
2. bisecting
3. measuring
4. rotating
Show answer
Answer: (a) tiling. It is also called tessellation.
Each interior angle of a square is
1. 60°
2. 90°
3. 108°
4. 120°
Show answer
Answer: (b) 90°. Four of them make 360° at a point.
A ray that divides an angle into two equal parts is the
1. perpendicular
2. angle bisector
3. diagonal
4. radius
Show answer
Answer: (b) angle bisector. Both parts are equal.
How many equilateral triangles meet at a point in a triangular tiling
1. 3
2. 4
3. 6
4. 8
Show answer
Answer: (c) 6. 6 × 60° = 360°.
The interior angle of a regular hexagon is
1. 108°
2. 120°
3. 135°
4. 144°
Show answer
Answer: (b) 120°. (6 − 2) × 180° ÷ 6 = 120°.
A perpendicular bisector of AB passes through
1. point A
2. the midpoint of AB at 90°
3. point B only
4. any point of AB
Show answer
Answer: (b) the midpoint of AB at 90°. It cuts AB into equal halves at a right angle.
Why can a regular pentagon not tile a plane by itself
1. its sides are unequal
2. it has no angles
3. it has too many sides
4. 360 is not a multiple of its angle 108°
Show answer
Answer: (d) 360 is not a multiple of its angle 108°. 3 × 108° = 324° leaves a gap, 4 × 108° overlaps.
Which combination can fill the space around a point exactly
1. two regular hexagons
2. three regular pentagons
3. two regular octagons and one square
4. one octagon and two triangles
Show answer
Answer: (c) two regular octagons and one square. 135° + 135° + 90° = 360°.

All chapters of Class 7 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.