Construction of Polygons
ICSE Class 9 · Mathematics · Chapter 14
What this chapter covers
- Ruler-and-compasses constructions
- Basic constructions you will use
- How many measurements fix a quadrilateral
- Quadrilaterals from sides and a diagonal
- Quadrilaterals from sides and angles
- Construction of a trapezium
- Construction of parallelograms
- Rectangles and squares
- Construction of a rhombus
- Construction of a regular hexagon
Key points
- Constructions in this chapter use only a ruler and a pair of compasses; all construction arcs are left visible.
- Angles of 60°, 120°, 90° and their bisections and combinations (30°, 45°, 75°, 105°, 135°, 150°) can be made without a protractor.
- A general quadrilateral needs five independent measurements; a diagonal splits it into two triangles, built one after the other.
- Always check the triangle inequality, and that the angle data are consistent, before drawing.
- A parallelogram needs three measurements; its fourth vertex is fixed by making opposite sides equal or by making the diagonals bisect each other.
- A rectangle needs two measurements and a square only one, using right angles and equal diagonals.
- A rhombus is fixed by a side and an angle, a side and a diagonal, or its two diagonals; its diagonals bisect each other at right angles.
- A trapezium uses the parallel-line construction; with four sides given, the hidden triangle is found by drawing a line through one vertex parallel to a non-parallel side.
- A regular hexagon has interior angles of 120° and side equal to the radius of its circumscribing circle, so it is drawn by stepping the radius six times round the circle.
Questions answered
Define a regular polygon, and state each interior angle of a regular hexagon.
A regular polygon is a polygon whose sides are all equal and whose interior angles are all equal. Each interior angle of a regular hexagon is ((6 - 2) × 180°)/(6) = 120°.
Why do four sides alone not fix a quadrilateral?
A frame of four fixed sides can be pushed out of shape, changing its angles while the sides stay the same. A fifth measurement, such as a diagonal or an angle, is needed to make it rigid.
Name the two instruments allowed in the constructions of this chapter, and explain how you would make an angle of 75° with them.
A ruler and a pair of compasses. Construct the 60° and 90° rays at the same point and bisect the angle between them; the bisector makes 60° + 15° = 75° with the base.
Can a quadrilateral ABCD be constructed with AB = 3 cm, BC = 3.5 cm, CD = 4 cm, DA = 4 cm and AC = 7 cm? Give a reason.
No. In triangle ABC, AB + BC = 6.5 cm, which is less than AC = 7 cm, so the triangle cannot be formed.
Construct a quadrilateral PQRS with PQ = 5 cm, QR = 4 cm, RS = 4.5 cm, SP = 3.5 cm and diagonal PR = 6 cm. Write the steps.
Draw PR = 6 cm. With centres P and R and radii 5 cm and 4 cm, cut arcs on one side of PR to get Q. With centres P and R and radii 3.5 cm and 4.5 cm, cut arcs on the other side to get S. Join PQ, QR, RS and SP.
In a parallelogram ABCD, AB = 6 cm, BC = 4 cm and ∠ B = 120°. What are the other three angles, and how is D fixed in the construction?
∠ D = 120° and ∠ A = ∠ C = 60°, because opposite angles are equal and adjacent angles add up to 180°. After drawing AB, ∠ B and BC, cut an arc of radius 4 cm from A and an arc of radius 6 cm from C; they meet at D.
The diagonals of a rhombus are 10 cm and 24 cm. Find its side, and describe how to construct it.
The half-diagonals are 5 cm and 12 cm and meet at right angles, so the side is √(52 + 122) = 13 cm. Draw AC = 24 cm, draw its perpendicular bisector through O, and mark B and D on it, 5 cm from O on either side; join the four points.
Explain why the perpendicular bisector is used in the construction of a square from its diagonal.
The diagonals of a square are equal and bisect each other at right angles. The perpendicular bisector of one diagonal passes through its mid-point at right angles, so the other two vertices lie on it, each half a diagonal from the mid-point.
Past board exam questions
ICSE 2019 sample paper · 3 marks
Construct a prallellogram ABCD in which AB = 6.4 cm, AD = 5.2 cm and the perpendicular distance between AB and DC is 4 cm.
Show answer
Steps of construction:
1. Draw AB = 6.4 cm.
2. At A draw AX ⊥ AB; cut AE = 4 cm on AX.
3. Through E draw line EY perpendicular to AE (i.e. parallel to AB).
4. With A as centre and radius 5.2 cm, draw an arc cutting EY at D.
5. With D as centre and radius 6.4 cm, cut EY at C (on the same side as B), or draw BC ∥ AD.
6. Join AD and BC. ABCD is the required parallelogram, with AB = 6.4 cm, AD = 5.2 cm and distance between AB and DC = 4 cm.