For the segment bounded by chord and arc , CD = 12 cm and m = 120 ° .Find the exact area of the segment.
A)48
B)(48 18 )
C)(48 36 )
D)None of These

Question

For the segment bounded by chord  and arc  , CD = 12  cm and m  = 120 ° .Find the exact area of the segment.
A)48 
B)(48 18 ) 
C)(48 36 ) 
D)None of These

Quizplus · Accepted Answer

The formula for the area of a segment is given by A = (1/2)r^2(θ-sinθ) where r is the radius of the circle and θ is the central angle in radians.

First, we need to find the radius of the circle. CD is a perpendicular bisector of the chord, so it divides the chord into 2 equal parts. Therefore, BD = 11/2 cm. In triangle BDC, we can use the Pythagorean theorem to find BD. Let x be the length of DC. Then, we have:

BD^2 + x^2 = 12^2
(11/2)^2 + x^2 = 144
x^2 = 144 - (121/4)
x = √(523/4)

Now, we can find the radius using the formula for the distance between the center of the circle and the midpoint of a chord:

r = √(2h^2 - d^2)/2
where h is the distance from the center to the chord and d is the length of the perpendicular from the center to the chord.

Since the chord and arc are symmetric about CD, the center must lie on CD. Let O be the center of the circle. Then, DO is the perpendicular from O to CD, so d = DO = BD - BO = 11/2 - r. Also, h = OD = r. Substituting these values into the formula, we get:

r = √(2r^2 - (11/2 - r)^2)/2
4r^2 = 2r^2 + (11/2 - r)^2
2r^2 - (11/2 - r)^2 = 0
3r^2 + 121/4 = 11r
3r^2 - 11r + 121/4 = 0
(3r - 11/2)^2 = 0
r = 11/6

Now, we can find the central angle θ using the formula:

θ = 2sin^-1(d/2r)
θ = 2sin^-1((11/4)/(11/3))
θ = 2sin^-1(3/4)
θ = 2(0.8481)
θ = 1.6962 radians

Finally, we can find the area of the segment using the formula:

A = (1/2)r^2(θ - sinθ)
A = (1/2)(11/6)^2(1.6962 - sin(1.6962))
A = 18.8557

Rounding to the nearest integer, we get:

A ≈ 19

Therefore, the best choice is C.

For the Segment Bounded by Chord and Arc