In a circle centered at point O, sector OAB has a central angle of 60^\circ and a radius o… — GRE General Test

In a circle centered at point $O$ , sector $OAB$ has a central angle of $60^\circ$ and a radius of $12$ . A smaller circle is inscribed inside sector $OAB$ such that it is tangent to radius $OA$ , radius $OB$ , and arc $AB$ . If the area of the region inside sector $OAB$ that lies outside the inscribed circle is expressed in the form $k\pi$ , what is the value of $k$ ?

Cevap: 8

Cevap

The correct value of

k

is 8.

By using the geometry of the

30^\circ

60^\circ

90^\circ

right triangle formed by the angle bisector and the radius of tangency, the radius of the inscribed circle is found to be

r = 4

. Subtracting its area (

16\pi

) from the sector's area (

24\pi

) gives

8\pi

, so

k = 8

Adım Adım Çözüm

Determine the relationship between the radius of the larger circle

R

and the radius of the inscribed circle

r

OP = 2r

and

R = 3r

The center

P

of the inscribed circle lies on the angle bisector of

\angle AOB = 60^\circ

, creating a

30^\circ

angle with radius

OA

. The perpendicular distance from

P

to radius

OA

r

, so

\sin(30^\circ) = \frac{r}{OP} = \frac{1}{2}

, giving

OP = 2r

. Since the inscribed circle touches arc

AB

OP + r = R

, so

3r = R

Calculate the radius

r

of the inscribed circle.

r = 4

Given

R = 12

, solving

3r = 12

yields

r = 4

Compute the area of sector

OAB

\text{Area}_{\text{sector}} = 24\pi

The formula for the area of a sector is

\frac{\theta}{360^\circ} \pi R^2

. Here,

\frac{60^\circ}{360^\circ} \pi (12^2) = \frac{1}{6} \times 144\pi = 24\pi

Compute the area of the inscribed circle.

\text{Area}_{\text{circle}} = 16\pi

The area of a circle with radius

r = 4

\pi r^2 = \pi (4^2) = 16\pi

Subtract the area of the inscribed circle from the area of sector

OAB

to find

k

k = 8

\text{Area}_{\text{region}} = 24\pi - 16\pi = 8\pi

, which means

k = 8

Anahtar Kavram

Inscribed shapes within sectors, arc length, and sector area relations

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