In fig. there are two concentric circles with Centre O of radii 5cm and 3cm. From an

external point P, tangents PA and PB are drawn to these circles if AP = 12cm, find the

tangent length of BP.

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#### Solution

OA = 5 cm

OB = 3 cm

AP = 12 cm

BP = ?

We know that

At the point of contact, radius is perpendicular to tangent.

For circle 1, ΔOAP is right triangle

By Pythagoras theorem, 𝑂𝑃^{2} = 𝑂𝐴^{2} + 𝐴𝑃^{2}

⇒ 𝑂𝑃^{2} = 5^{2} + 12^{2} = 25 + 144

= 169

⇒ OP = `sqrt(169)` = 13 𝑐𝑚

For circle 2, ΔOBP is right triangle by Pythagoras theorem,

𝑂𝑃^{2} = 𝑂𝐵^{2} + 𝐵𝑃^{2}

13^{2} = 3^{2} + 𝐵𝑃^{2}

𝐵𝑃^{2} = 169 − 9 = 160

𝐵𝑃 = `sqrt(160) = 4sqrt(10)` 𝑐𝑚

Concept: Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles

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