In the given figure, PA and PB are two tangents drawn from an external point P to a circle with center C and radius 4 cm. If PA ⏊ PB, then the length of each tangent is

Join AC, BC and CP

To Find: Length of tangents

Now,

PA = PB…[1]

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

In △ACP and △BCP

PA = PB [By 1]

CP = CP [Common]

CA = CB [radii of same circle]

△ACP ≅△BCP [By Side - Side - Side Criterion]

∠CPA = ∠CPB

[Corresponding parts of congruent triangles are congruent]

Now,

∠APB = 90°

[Given that PA ⏊ PB]

∠CPA + ∠CPB = 90°

∠CPA + ∠CPA = 90°

2 ∠CPA = 90°

∠CPA = 45°

In △ACP

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

∴ ACP is a right - angled triangle.

So, we have

⟹PA = 4 cm

From [1]

PA = PB = 4 cm

i.e. length of each tangent is 4 cm

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