If two tangents inclined at an angle of 60° are drawn to a circle of radius 3 cm then the length of each tangent is

Let us consider a circle with center O and AP and BP are two tangents such that angle of inclination i.e. ∠APB = 60°

Joined OA, OB and OP.

To Find : Length of tangents

Now,

PA = PB [Tangents drawn from an external point are equal] [1]

In △AOP and △BOP

PA = PB [By 1]

OP = OP [Common]

OA = OB [radii of same circle]

△AOP ≅△BOP

[By Side - Side - Side Criterion]

∠OPA = ∠OPB

[Corresponding parts of congruent triangles are congruent]

Now,

∠APB = 60° [Given]

∠OPA + ∠OPB = 60°

∠OPA + ∠OPA = 60°

2 ∠OPA = 60°

∠OPA = 30°

In △AOP

OA ⏊ PA

[Tangents drawn at a point on circle is perpendicular to the radius through point of contact

∴ AOP is a right - angled triangle.

So, we have

⟹PA = 3√3 cm

From [1]

PA = PB = 4 cm

i.e. length of each tangent is 3√3 cm

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