Question

Draw a circle of diameter 12 cm. From a point 10 cm away from its centre, construct a pair of tangents to the circle. Measure the length of each tangent segment.

Answer 1

Given that tangents PQ and PR are drawn from a point P to a circle with centre O.
Here, OQ = OR = 6 cm and OP = 10 cm.


Now,
PQ is a tangent at Q and OQ is the radius through Q.
∴ OQ ⊥ QP
Similarly,
OR ⊥ PR
In the right-angled ΔOQP and ΔORP, we get:
OQ = OR [ radii of the same circle]
OP = OP [ common side]
∴ ΔOQP ≅ ΔORP [ By RHS congruence]
Hence QP = PR
Again,
By Pythagoras' theorem, we get:
⇒ OP² = OQ² + QP²
⇒ (10)² = (6)² + QP²
⇒ QP² = [100 - 36] = 64 ⇒ QP = 8 cm
Thus, QP = RP = 8 cm