Question

In the following figure, shows a sector of a circle, centre O, containing an angle θ°. Prove that:
(i)Perimeter of the shaded region is $(tan\Theta+sec\Theta+\frac{\pi\Theta}{180}-1)$
(ii)Area of the shaded region is $\frac{r^{2}}{2} (tan\Theta-\frac{\pi\Theta}{180})$

Answer 1

Answer:

Proved , Perimeter of shaded region = r(tan θ  + secθ + πθ /180 - 1)  and Area of a shaded  region = r²/2 ( tan θ - π θ/180)

Step-by-step explanation:

Figure of this question is in the attachment.  

Given :

Angle subtended at the centre of a circle = θ

Length of Arc AC = θ /360 × 2πr  

Length of Arc AC= θπ r/180

tan θ = P/B = AB /OA  

tan θ = AB /r  

AB = r tan θ  

cos θ = B/H = OA /OB  

cos θ = r /OB  

OB = r /cos θ = r × 1/cos θ

OB = r × sec θ

[sec θ = 1/cos θ]

OB = r sec θ

BC = OB - OC  

BC = r sec θ -r  

BC = r(sec θ -1)

(i) Perimeter of shaded region = AB+ BC + arc AC  

= r tanθ θ +  r secθ - r  + θπ r/180

= r(tan θ  + secθ - 1 + πθ /180)

Perimeter of shaded region = r(tan θ  + secθ + πθ /180 - 1)

Hence, Perimeter of shaded region = r(tan θ  + secθ + πθ /180 - 1)

(ii) Area of a shaded  region, A =  Area of right ∆ OAB - Area of  

sector  

A = 1/2 ×OA × AB - θ/360 ×πr²

A= r × r tanθ /2 - πr² × θ /360

A = r²/2 ( tan θ - π θ/180)

Hence, Area of a shaded  region = r²/2 ( tan θ - π θ/180)

HOPE THIS ANSWER WILL HELP YOU….

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AB is a chord of a circle with centre O and radius 4 cm. AB is of length 4 cm. Find the area of the sector of the circle formed by chord AB.

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Answer 2

Step-by-step explanation:

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