Question

Draw a circle of radius 3 cm. Take any point K on it. Draw a tangent to the circle from point K without using center of the circle.

Answer 1

Analysis:

As shown in the figure, line l is a tangent to the circle at point K.

seg BK is a chord of the circle and ∠BAK is an inscribed angle.

By tangent secant angle theorem,

∠BAK = ∠BKR

By converse of tangent secant angle theorem,

If we draw ∠BKR such that ∠BKR = ∠BAK, then ray KR

i.e. (line l) is a tangent at point K.

Steps of construction:

1. Draw a circle of radius 3 cm and take any point K on it.
2. Draw chord BK of any length and an inscribed ∠BAK of any measure.
3. By taking A as a centre and any convenient distance on the compass draw an arc intersecting the arms of ∠BAK in points P and Q.
4. With K as a centre and the same distance in the compass, draw an arc intersecting the chord BK at point S.
5. Taking radius equal to PQ and S as the centre, draw an arc intersecting the previously drawn arc. Name the point of intersection as R.
6. Draw line RK. Line RK is the required tangent to the circle.