Draw a circle with the help of a bangle. Take a point outside the circle. Construct the pair of tangents from this point to the circle.

Steps of construction:

Draw any circle using a bangle.

To find its center:

• Draw two chords on the circle say AB and CD.
• Draw the perpendicular bisectors of AB and CD to intersect at O.

Now, ‘O’ is the center of the circle (since the perpendiculars drawn from the center of a circle to any chord bisect the chord and vice versa).

To draw the tangents from a point ‘P’ outside the circle:

• Take a point P outside the circle and draw the perpendicular bisector of OP which meets at OP at O’.
• With O’ as the center and OO’ as radius draw a circle that cuts the given circle at Q and R.
• Join PQ and PR.

Thus, PQ and PR are the required tangents.

Proof:

∠QOP = ∠ORP = 90° (Angle in a semi-circle)

∴ OQ ⊥ QP and OR ⊥ RP. (We know that the line joining the center of a circle to the tangent is always perpendicular)

Hence, we have PQ and PR as the tangents to the given circle.