Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths. Give the justification of the construction.
A pair of tangents to the given circle can be constructed as follows.
Taking any point O of the given plane as centre, draw a circle of 6 cm radius. Locate a point P, 10 cm away from O. Join OP.
Bisect OP. Let M be the mid-point of PO.
Taking M as centre and MO as radius, draw a circle.
Let this circle intersect the previous circle at point Q and R.
Join PQ and PR. PQ and PR are the required tangents.
The lengths of tangents PQ and PR are 8 cm each.
The construction can be justified by proving that PQ and PR are the tangents to the circle (whose centre is O and radius is 6 cm). For this, join OQ and OR.
∠PQO is an angle in the semi-circle. We know that angle in a semi-circle is a right angle.
∴ ∠PQO = 90°
⇒ OQ ⊥ PQ
Since OQ is the radius of the circle, PQ has to be a tangent of the circle. Similarly, PR is a tangent of the circle.