Question

In a circular museum hall of radius 14 m, some statues are displayed. Statues are kept inside the inner concentric circle of radius 7 m. One such statue lying in sector OAB, is fenced along line segments OA, AP, PB and BO where P is a point on outer circle.

Based on above information, answer the following questions:

(i) Find m∠AOP.   [1]

(ii) Prove that ΔOAР ≅ ΔОВР.   [1]

(iii) (a) Find the length of fencing required to protect the statue. (Take `sqrt(3) = 1.73`)   [2]

OR

(b) Find area of quadrilateral OAPB. (Take `sqrt(3) = 1.73`)

Answer 1

(i) Let ∠AOP = θ

`cos θ = B/H = (AO)/(OP)`

`cos θ = 7/14`

`cos θ = 1/2`

cos θ = cos 60°

θ = 60°

(ii) In ΔOAР and ΔОВР

OA = OB   ...(Radius)

OP = OP   ...(Common)

∠OAP = ∠OBP = 90°   ...(Tangent ⊥ Radius)

By RHS congruency

ΔOAР ≅ ΔОВР

(iii) (a) Length of fencing = OA + OB + BP + AP

= 7 + 7 + BP + AP

Now, By Pythagoras theorem:

(PO)² = (PA)² + (AO)²

(14)² = (PA)² + (7)²

196 – 49 = AP²

`sqrt(147) = AP`

`AP = 7sqrt(3)`

= 7 × 1.73

= 12.11 m

OR

(iii) (b) Area of quadrilateral OAPB = `2 xx 1/2 xx OA xx AP`

= `7 xx 7sqrt(3)`

= `49sqrt(3)`

= 49 × 1.73

= 84.77 m²