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CBSE · Class 7 · Mathematics

Chapter 14: Constructions and Tilings

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी · हिंदी

Bisecting segments and angles

From daily life

Folding paper so that two points land on each other makes a crease: that crease is the perpendicular bisector of the segment joining them.

Simple definition

A perpendicular bisector cuts a segment into two equal parts at 90°; each of its points is equally far from both ends. An angle bisector splits an angle into two equal angles. Both are drawn with a compass and ruler.

ABMPA = PB

In the figure: Every point P on the perpendicular bisector has PA = PB.

Live figure

Bisecting segments and angles

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40°40°bisector80° → 40° + 40°

In the figure: An angle bisector halves the angle.

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Bisecting segments and angles

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Rule 1: Perpendicular bisector steps — Open the compass more than half of AB; draw arcs from A and B; join the two crossing points.

  • Easy: AB = 10 cm → AM = 5 cm.
  • Medium: P on the bisector with PA = 6 → PB = 6.
  • Exam-type: P with PA = 4 and PB = 9 is not on it.

Rule 2: Angle bisector steps — Arc across both arms; from those two points draw equal arcs that cross; join the vertex to the crossing.

  • Easy: Bisect 100° → 50° each.
  • Medium: Bisect 90° then one half again → 22.5°.
  • Exam-type: Bisect 70° → 35° each.

Rule 3: Circumcentre — Perpendicular bisectors of the three sides meet at a point equidistant from all vertices.

  • Easy: Draw two bisectors; their crossing is the centre.
  • Medium: A circle around the triangle is centred there.
  • Exam-type: Two bisectors are enough; the third passes through the same point.
Remember

Perpendicular bisector: 90° and midpoint.

Common mistake

Opening the compass to exactly half of AB. It must be more than half.

Shortcut

Keep the same compass width for both arcs. Memory clue: "Same width, cross arcs."

Spot the difference

BisectorSplits
Perpendicular bisectorsegment at 90°
Angle bisectorangle into two equal

Practice questions

Easy
  1. 1AB = 10 cm, perpendicular bisector meets AB at M. AM (cm)?
  2. 2P is on the perpendicular bisector of AB, PA = 6 cm. PB (cm)?
  3. 3Bisect 100°. Each part (degrees)?
Medium
  1. 4Bisect 70°. Each part (degrees)?
  2. 5Bisect 90°, then bisect one half. Smallest angle?
  3. 6Compass opening for bisecting AB = 8 cm must be more than (cm)?
Challenging
  1. 7Point with PA = 4 and PB = 9: on the perpendicular bisector?
  2. 8Perpendicular bisectors of a triangle's sides meet at the…
Show answers (try them yourself first)
  1. 15
  2. 26
  3. 350
  4. 435
  5. 522.5
  6. 64
  7. 7No
  8. 8circumcentre

Constructing triangles

From daily life

Three sticks of 3, 4 and 5 cm can be joined into a triangle, but sticks of 5, 6 and 12 cm cannot: the two short ones cannot reach each other.

Simple definition

A triangle can be built when any two sides add up to more than the third. A triangle is fixed by SSS, SAS or ASA measurements; AAA fixes only the shape. The three angles add up to 180°.

5343 + 4 > 5 ✔5 + 6 < 12 ✘

In the figure: Sides 3, 4, 5 work; sides 5, 6, 12 do not.

Live figure

Constructing triangles

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Rule 1: Triangle inequality test — Add the two smaller sides; compare with the largest.

  • Easy: 3, 4, 5: 3 + 4 = 7 > 5 → possible.
  • Medium: 5, 6, 12: 5 + 6 = 11 < 12 → impossible.
  • Exam-type: 4, 4, 9: 4 + 4 = 8 < 9 → impossible.

Rule 2: Range of the third side — Third side lies between the difference and the sum of the other two.

  • Easy: Sides 5 and 8: third between 3 and 13 → 9 works.
  • Medium: Sides 6 and 10: whole numbers from 5 to 15.
  • Exam-type: Sides 6 and 9: largest whole third side is 14.

Rule 3: What fixes a triangle — SSS, SAS, ASA fix one triangle; AAA does not.

  • Easy: Sides 6, 7, 8 give exactly one triangle.
  • Medium: Angles 50° and 60° leave the third angle 70°.
  • Exam-type: Angle ratio 1 : 2 : 3 gives 30°, 60°, 90°.
Remember

Two sides together must beat the third.

Common mistake

Testing with the two longest sides. Test the two smallest against the largest.

Shortcut

Sum of the two small sides must exceed the biggest. Memory clue: "Short two beat the long one."

Spot the difference

GivenUnique triangle?
SSS, SAS, ASAYes
AAANo

Practice questions

Easy
  1. 1Can sides 5, 7, 9 form a triangle?
  2. 2Can sides 4, 4, 9 form a triangle?
  3. 3Two angles are 35° and 65°. The third (degrees)?
Medium
  1. 4Two sides 6 and 10. Smallest whole third side?
  2. 5Two sides 6 and 10. Largest whole third side?
  3. 6Do three given angles fix a unique triangle?
Challenging
  1. 7Angles are in the ratio 2 : 3 : 4. The largest (degrees)?
  2. 8Angles are in the ratio 1 : 2 : 3. The largest (degrees)?
Show answers (try them yourself first)
  1. 1Yes
  2. 2No
  3. 380
  4. 45
  5. 515
  6. 6No
  7. 780
  8. 890

Constructing special angles

From daily life

With only a compass and a ruler, no protractor, carpenters still mark exact 60°, 90° and 45° corners.

Simple definition

A compass makes 60° (arcs of the same radius give an equilateral triangle) and 90° (perpendicular). By bisecting or adding angles you get 30°, 45°, 120°, 135° and more.

30°60°60° ÷ 2 = 30°

In the figure: Bisecting the 60° angle gives 30°.

Rule 1: Halve to get smaller — 60° → 30°; 90° → 45°; 45° → 22.5°; 30° → 15°.

  • Easy: Bisect 60° → 30°.
  • Medium: Bisect 90° → 45°.
  • Exam-type: Bisect 30° → 15° (two bisections of 60°).

Rule 2: Add to get larger — Put two constructed angles side by side.

  • Easy: 60° + 60° = 120°.
  • Medium: 90° + 45° = 135°.
  • Exam-type: 60° + 15° = 75°; 90° + 15° = 105°.

Rule 3: Subtract from a straight angle — 180° minus a built angle gives its supplement.

  • Easy: 180° – 30° = 150°.
  • Medium: 180° – 15° = 165°.
  • Exam-type: 180° – 60° = 120°.
Remember

Special angles come from 60° and 90° by halving and adding.

Common mistake

Thinking 50° can be built exactly this way. Only multiples of 7.5° appear.

Shortcut

Plan the angle as sums and halves before you draw. Memory clue: "60 and 90, halve and add."

Spot the difference

AngleHow
30°bisect 60°
75°60° + 15°
135°90° + 45°

Practice questions

Easy
  1. 1Which shape gives a 60° angle with a compass?
  2. 2Bisect 60°. Angle (degrees)?
  3. 3Bisect 90°. Angle (degrees)?
Medium
  1. 460° + 60° =
  2. 575° = 60° + □. □ =
  3. 6135° = 90° + □. □ =
Challenging
  1. 722.5° is half of (degrees)?
  2. 8165° = 180° – □. □ =
Show answers (try them yourself first)
  1. 1an equilateral triangle
  2. 230
  3. 345
  4. 4120
  5. 515
  6. 645
  7. 745
  8. 815

Regular polygons and tiling

From daily life

Bathroom floors use identical tiles with no gaps and no overlaps. Only some shapes can do that.

Simple definition

A regular polygon has equal sides and equal angles. Its interior angle is (n – 2) × 180° ÷ n and its exterior angle is 360° ÷ n. Identical regular polygons tile a floor only if their angles fit exactly 360° around each corner.

3 × 120° = 360°

In the figure: Three hexagons at each corner: 3 × 120° = 360°.

Rule 1: Interior angles — Triangle 60°, square 90°, pentagon 108°, hexagon 120°, octagon 135°, decagon 144°, dodecagon 150°.

  • Easy: Hexagon: 4 × 180° ÷ 6 = 120°.
  • Medium: Octagon: 6 × 180° ÷ 8 = 135°.
  • Exam-type: 12 sides: 10 × 180° ÷ 12 = 150°.

Rule 2: Exterior angle — 360° ÷ number of sides; interior + exterior = 180°.

  • Easy: Hexagon exterior = 360° ÷ 6 = 60°.
  • Medium: Exterior 36° → 10 sides.
  • Exam-type: Exterior 40° → 9 sides.

Rule 3: Which shapes tile alone? — 360° must divide evenly by the interior angle.

  • Easy: Triangles: 360 ÷ 60 = 6 meet at a corner.
  • Medium: Squares: 4 meet; hexagons: 3 meet.
  • Exam-type: Pentagon: 360 ÷ 108 is not whole → no.
Remember

Tiling needs the angle to divide 360° exactly.

Common mistake

Believing pentagons tile. 108° does not divide 360°.

Shortcut

Only triangle, square and hexagon tile by themselves. Memory clue: "3, 4, 6 sides tile alone."

Spot the difference

ShapeInteriorTiles alone?
Triangle60°Yes
Hexagon120°Yes
Pentagon108°No

Practice questions

Easy
  1. 1Interior angle of a regular pentagon (degrees)?
  2. 2Interior angle of a regular hexagon (degrees)?
  3. 3Interior angle of a regular octagon (degrees)?
Medium
  1. 4Interior angle of a regular 12-sided polygon (degrees)?
  2. 5Exterior angle of a regular hexagon (degrees)?
  3. 6Exterior angle 36°. Number of sides?
Challenging
  1. 7How many regular hexagons meet at a corner in a tiling?
  2. 8Can regular pentagons alone tile a floor?
Show answers (try them yourself first)
  1. 1108
  2. 2120
  3. 3135
  4. 4150
  5. 560
  6. 610
  7. 73
  8. 8No

Tiling with combinations

From daily life

Many old floors mix two shapes, like octagons with small squares. Around every corner the angles still add to exactly 360°.

Simple definition

Different regular polygons can tile together if the interior angles at a corner add up to exactly 360°. Add the angles you have, then subtract from 360° to see what is still missing.

90° × 4 = 360°

In the figure: Four 90° corners fill the full turn of 360°.

Rule 1: Check the corner sum — Add the interior angles of all shapes at one corner.

  • Easy: 4 squares: 4 × 90° = 360°.
  • Medium: 3 triangles + 2 squares: 180° + 180° = 360°.
  • Exam-type: 1 square + 2 octagons: 90° + 270° = 360°.

Rule 2: Find the missing angle — 360° minus the angles already placed.

  • Easy: 2 hexagons use 240°, leaving 120° → a third hexagon.
  • Medium: 2 pentagons use 216°, leaving 144° → a decagon.
  • Exam-type: 2 dodecagons use 300°, leaving 60° → a triangle.

Rule 3: Find the polygon from its angle — Match the missing angle with the list of interior angles.

  • Easy: Missing 90° → square; 120° → hexagon.
  • Medium: Interior angle 135° → 8 sides.
  • Exam-type: Hexagon + square + dodecagon: 120° + 90° + 150° = 360°.
Remember

Around one corner the angles must total 360°.

Common mistake

Stopping at 350° or going over 360°. There must be no gap and no overlap.

Shortcut

Subtract the placed angles from 360° and look up the leftover. Memory clue: "Sum to 360, no gap, no overlap."

Spot the difference

ResultMeaning
< 360°gap
> 360°overlap
= 360°perfect fit

Practice questions

Easy
  1. 1Total of 4 square corners at a point (degrees)?
  2. 2Two hexagons at a corner leave how many degrees?
  3. 33 triangles + 2 squares: total at a corner (degrees)?
Medium
  1. 41 square + 2 octagons: total (degrees)?
  2. 51 triangle + 2 dodecagons: total (degrees)?
  3. 6Hexagon + square + dodecagon: total (degrees)?
Challenging
  1. 7Two pentagons leave how many degrees?
  2. 8Do 2 pentagons and 1 decagon fit at a corner?
Show answers (try them yourself first)
  1. 1360
  2. 2120
  3. 3360
  4. 4360
  5. 5360
  6. 6360
  7. 7144
  8. 8Yes