ICSE Class 8 Maths Geometry Chapter 05 Constructions

Constructions

In this chapter, we will revise some basic constructions that you learnt in the previous classes. You will also learn how to construct quadrilaterals, circumcircles and incircles. You will need only a ruler and a compass.

Basic Constructions

To construct an angle equal to a given angle

Construction 1 Construct an angle equal to \(\angle ABC\) at the point X.

Steps of construction

1. With B as the centre and a suitable radius, draw an arc to intersect BA at D and BC at E.

2. Draw a line XY through X.

3. With X as the centre and the same radius (as in Step 1), draw an arc which intersects XY at P.

4. With P as the centre and a radius equal to DE, draw an arc which intersects the arc drawn in Step 3 at Q.

5. Join X and Q and extend it to Z. Then, \(\angle YXZ\) is the required angle.

To bisect a given angle

Construction 2 Bisect the given angle ABC.

Steps of construction

1. With B as the centre and a convenient radius, construct an arc which intersects BA at P and BC at Q.

2. With P as the centre and a suitable radius (more than half of PQ), draw an arc.

3. With Q as the centre and the same radius, draw another arc to intersect the arc drawn in Step 2 at O.

4. Join B and O and extend it to P. Then, the ray BP bisects \(\angle ABC\).

To construct angles of standard magnitudes

Construction 3 Construct an angle of 60°.

Steps of construction

1. Draw a line OP.

2. With O as the centre and a suitable radius, draw an arc to intersect OP at A.

3. Draw another arc of the same radius with A as the centre. Let it intersect the arc drawn in Step 2 at the point B.

4. Draw a ling joining O and B and extend it to Q. Then, \(\angle POQ\) is the required angle of 60°.

Construction 4 Construct an angle of 30°.

Steps of construction

1. Draw \(\angle POQ\) of 60° as in the previous construction.

2. Bisect the \(\angle POQ\) as in Construction 2.

3. If OR is the bisector of \(\angle POQ\) then \(\angle POR = \angle QOR = 30°\).

Construction 5 Construct an angle of 120°.

Steps of construction

1. Draw a line OP.

2. Draw an arc with O as the centre and a convenient radius to intersect OP at the point A.

3. Draw another arc of the same radius with A as the centre. Let it intersect the arc drawn in Step 2 at the point B.

4. Draw a third arc of the same radius with B as the centre so that it cuts the arc drawn in Step 2 at the point C (away from A).

5. Draw a line joining O and C and extend it to Q. \(\angle QOP\) is an angle of measure 120°.

Construction 6 Construct an angle of 90°.

Steps of construction

Follow the Steps 1 to 4 in the previous construction.

5. Draw an arc of a suitable radius (more than \(\frac{1}{2} BC\)) with B as the centre.

6. Draw another arc of the same radius with C as the centre. Let this arc intersect the arc drawn in Step 5 at the point Q.

7. Draw a line through O and Q. Then \(\angle POQ = 90°\).

Construction 7 Construct an angle of 45°.

Steps of construction

1. Follow the Steps in Construction 6 to draw \(\angle POQ = 90°\).

2. Bisect \(\angle POQ\) (see Construction 2). If OR is the bisector then \(\angle POR = \angle QOR = 45°\).

To construct perpendicular lines

Construction 8 Construct a perpendicular to a line AB from an external point O.

Steps of construction

1. Draw a line AB.

2. Mark an external point O.

3. With O as the centre and a sufficiently large radius, draw an arc intersecting AB at the points P and Q.

4. With P as the centre and a radius more than \(\frac{1}{2} PQ\) (\(\neq OP\)), draw another arc on any side of AB.

5. With Q as the centre and the same radius, draw a third arc, intersecting the arc drawn in Step 4 at the point R.

6. Draw a line joining O and R. Let it intersect AB at the point S. Then OS is the required perpendicular.

Construction 9 Construct a line perpendicular to a line AB through a point O on the line.

Steps of construction

1. Draw a line AB.

2. Take a point O on AB.

3. Draw an arc with O as the centre and a convenient radius. Let it intersect the line AB at the points P and Q.

4. Draw an arc with P as the centre and a radius more than \(\frac{1}{2} PQ\).

5. With Q as the centre and the same radius, construct another arc to intersect the arc drawn in Step 4 at the point R.

6. The line joining O and R is the perpendicular to the line AB at O.

To construct the perpendicular bisector of a line segment

Construction 10 Construct the perpendicular bisector of a line segment AB.

Steps of construction

1. Draw a line segment AB.

2. Draw two arcs with A as the centre and a radius more than \(\frac{1}{2} AB\), one on each side of AB.

3. With B as the centre and the same radius, construct two more arcs, one on each side of AB to intersect the arcs drawn in Step 2 at the points P and Q respectively.

4. The line joining P and Q is the perpendicular bisector of AB. Let it intersect AB at O. Then, OA = OB = \(\frac{1}{2} AB\) and \(\angle AOP = \angle BOP = 90°\).

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