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Question

Draw a circle of radius 6 cm. From a point 10 cm away from its centre construct a pair of tangents to the circle. Measure the length of each of the tangent segments.

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Solution

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 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]
∴ Δ OQP ≅ Δ ORP [ 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

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