Question

The diagram shows a sector of a circle, OMN. The angle MON is 2x radians with radius r and center O.If perimeter P=20, what is value of Area A in terms of X?

Answer 1

Given : A sector of triangle is given with angle 2x radian & Perimeter

To find :  Area of Sector

Solution:

if perimeter is of Complete sector

Perimeter P  = r + r  +  (2x/2π) 2πr

=> P = 2r + 2rx

=> P = 2r(1 + x)

2r(1 + x)  = 20

=> r = 10/(1+ x)

Area of sector   =  ( 2x/2π) πr²

=  xr²

= x (10/(1 + x))²

= 100x/(1 + x)²

if perimeter is of arc :

Perimeter P  = (2x/2π) 2πr

=> P =  2rx

2rx  = 20

=> r = 10/x

Area of sector   =  ( 2x/2π) πr²

=  xr²

= x (10/x))²

= 100x/ x ²

= 100/x

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Answer 2

Given: Sector of a circle OMN,  MON is 2x radians, P=20

To find: Area A in terms of x?

Solution:

• Now, perimeter = 20

        OM + MN + NO = 20

        2r + MN = 20

• Expression for length of a circular arc with radius r and angle 2x rad is:

          s = r(theta)

         MN = r(2x)

         20 - 2r = 2rx

         20 = 2rx + 2r

         10 = rx + r

          r² = (10/x+1)²

• Divide by r, we get:

         10 / r = x + 1

          x = 10 / r - 1

• Now area  = 1/2 r²(theta)

            = 1/2 (r²)(2(x))

            = r²x

            =  (10/x+1)² x

Answer:

                So the Area A in terms of x is  (10/x+1)² x