To draw Tangent to a given circle at any point on it

 1) With centre ‘O’, draw a circle of any radius, and mark a point ‘P’ on it.

2) Draw a line joining O and P.

3)  Extend this line OP to Q so that OP=PQ.

4) Take any convenient radius and O as centre draw an arc as shown below,

5)  By taking same radius and Q as centre draw another arc cutting previously drawn arc as shown

6) Draw a line through P and R. Then this line is the required Tangent.

Question

1) Construct a tangent at any point P on a circle of radius 3 cm.

2) Draw a circle of radius 4 cm and construct a tangent at any point on the circle.

 

Constructing the perpendicular from a point to a line

           When two lines intersect to form right angles then such lines are known as Perpendicular to each other. Perpendicular lines are Co-Planar (i.e., They lie in same plane) and intersect at right angle (90°).

Method 1: Constructing a perpendicular to a line through a point on it

1) Assume given line as AB and a point P is present near middle of the line on it,

2) Taking P as a center and any suitable radius on compass, draw an arc cutting line segment AB at two distinct points (C, D) as shown below

3) Taking C as centre and a suitable radius (more than CP or DP distance) draw an arc above the line segment, with D as centre and the same radius as previous, draw an arc cutting previously drawn arc,

4) Mark the intersecting point as E.

5) Join the points E and P as shown, and the line segment EP is the required perpendicular to line AB through point P.

Note: This method we follow when we don't have Protractor with us.

Bisecting an Angle

           An angle bisector divides an angle into equal angles. If the angle is 'θ', the two angles made after bisecting will be 'θ/2'. Angle bisectors can be constructed for an acute angle, obtuse angle, or a right angle too. This angle bisector passes through the vortex of an angle, as shown in the figure.

Problem: To bisect a given Angle ABC

1. Let us consider some angle ∠ABC, B as vortex,

2. Take some length in compass and place your compass on the vertex of the angle, here on point B. Draw an arc that crosses both sides of the angle, lines AB and BC. Mark those points as D,E as shown,

3. Place your compass on one of the intersection points (i.e., D) and draw an arc inside the angle. Do not adjust the compass,

4. Place your compass on the other intersection point (i.e., E) and draw an arc that cuts the previously drawn arc as shown,

5. Mark the intersecting point of arc's as F. Finally, draw a line connecting the vortex B and point F. This is the angular bisector of our angle ∠ABC.

6. You can measure both sides of the angle to check they are equal.