Notes On Basic Constructions - CBSE Class 9 Maths
Geometrical instruments are used in drawing geometric figures such as triangles, circles, quadrilaterals, polygons etc. with given measuremets. A geometrical construction is the method of drawing a geometrical figure using an ungraduated ruler and a compass. An angle bisector is a ray, which divides an angle in to two equal parts. The bisector of a line segment is a line that cuts the line segment into two equal halves. A perpendicular bisector is a line, which divides a given line segment into two equal halves and is also perpendicular to the line segment.   Construction of the bisector of a given angle: Consider ∠DEF to construct the bisector. Steps of construction: Step 1: With E as centre and small radius draw arcs on the rays ED and EF. Step 2: Let the arcs intersect the rays ED and EF at G and H respectively. Step 3: With centres G and H, draw two more arcs with the same radius such that they intersect at a point. Let the point of intersection be I. Step 4: Draw a ray with E as the starting point and passing through I. EI is the bisector of the ∠DEF.  Construction of the perpendicular bisector of a line segment: Consider the line segment PQ to construct the perpendicular bisector. Steps of Construction: Step 1: Draw a line segment PQ. Step 2: With P as centre, draw two arcs on either sides of PQ with radius more the half the length of the given line segment. Step 3: Similarly draw two more arcs with same radius from point Q such that they intersect the previous arcs at R and S respectively. Step 4: Join the points R and S.  RS is the required perpendicular bisector of the given line segment PQ.                                Construction of an angle of 60° at the initial point of a given ray. Consider ray PQ with P as the initial point. Construction of a ray PR such that it makes angle of 60° with PQ. Steps of Construction: Step 1: Draw a ray PQ. Step 2: With P as centre, draw an arc with small radius such that it intersects the ray PQ at C. Step 3: With C as centre and same radius draw another arc to intersect the previous arc at D. Step 4: Draw a ray PR from point P through D. Hence, ∠RPQ is equal to 60°.

#### Summary

Geometrical instruments are used in drawing geometric figures such as triangles, circles, quadrilaterals, polygons etc. with given measuremets. A geometrical construction is the method of drawing a geometrical figure using an ungraduated ruler and a compass. An angle bisector is a ray, which divides an angle in to two equal parts. The bisector of a line segment is a line that cuts the line segment into two equal halves. A perpendicular bisector is a line, which divides a given line segment into two equal halves and is also perpendicular to the line segment.   Construction of the bisector of a given angle: Consider ∠DEF to construct the bisector. Steps of construction: Step 1: With E as centre and small radius draw arcs on the rays ED and EF. Step 2: Let the arcs intersect the rays ED and EF at G and H respectively. Step 3: With centres G and H, draw two more arcs with the same radius such that they intersect at a point. Let the point of intersection be I. Step 4: Draw a ray with E as the starting point and passing through I. EI is the bisector of the ∠DEF.  Construction of the perpendicular bisector of a line segment: Consider the line segment PQ to construct the perpendicular bisector. Steps of Construction: Step 1: Draw a line segment PQ. Step 2: With P as centre, draw two arcs on either sides of PQ with radius more the half the length of the given line segment. Step 3: Similarly draw two more arcs with same radius from point Q such that they intersect the previous arcs at R and S respectively. Step 4: Join the points R and S.  RS is the required perpendicular bisector of the given line segment PQ.                                Construction of an angle of 60° at the initial point of a given ray. Consider ray PQ with P as the initial point. Construction of a ray PR such that it makes angle of 60° with PQ. Steps of Construction: Step 1: Draw a ray PQ. Step 2: With P as centre, draw an arc with small radius such that it intersects the ray PQ at C. Step 3: With C as centre and same radius draw another arc to intersect the previous arc at D. Step 4: Draw a ray PR from point P through D. Hence, ∠RPQ is equal to 60°.

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