Q50 of 52 Page 10

In Fig. 9, is shown a sector OAP of a circle with center O, containing ∠θ AB is perpendicular to the radius OA and meets OP produced at B. Prove that the perimeter of shaded region is r

(CBSE 2016)

The length of the arc AP is given as:


Now, ΔOAB is a right angle triangle (∵ the tangent is perpendicular to the radius through the point of contact)


Now,


⇒ AB = OA× tanθ


⇒ AB = r× tanθ


Also, in right Δ OAB,



⇒ OB = OA × secθ


⇒ OB = r× secθ


Now, PB = OB-OP


⇒ PB = r.secθ – r


∴ The perimeter of the shaded area = Arc AP + AB + PB


= + r. tanθ + r.secθ – r


=

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