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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