Answer:
Radius of the circle: [tex]18\; {\rm cm}[/tex].
Angle subtended at the center of this circle: [tex](2/3)[/tex] radians, which is approximately [tex]38.2^{\circ}[/tex].
Step-by-step explanation:
Assume that the radius of the circle is [tex]r\; {\rm cm}[/tex], and the angle subtended at the center of the circle is [tex]\theta[/tex] radians. A full revolution is [tex]2\, \pi[/tex] radians, and this sector would only account for a fraction of [tex](\theta / (2\,\pi))[/tex] of the full circle:
Given that the arc length is [tex]12\; {\rm cm}[/tex], while the area of this sector is [tex]108\; {\rm cm^{2}}[/tex]:
[tex]\displaystyle \left(\frac{\theta}{2\, \pi}\right)\, 2\, \pi\, r = 12[/tex].
[tex]\displaystyle \left(\frac{\theta}{2\, \pi}\right)\, \pi\, r^{2} = 108[/tex].
One possible way to find [tex]r[/tex] is to eliminate [tex]\theta[/tex]. To do so, divide the second equation with the first:
[tex]\displaystyle \frac{\displaystyle \left(\frac{\theta}{2\, \pi}\right)\, \pi\, r^{2}}{\displaystyle \left(\frac{\theta}{2\, \pi}\right)\, 2\, \pi\, r} = \frac{108}{12}[/tex].
[tex]\displaystyle \frac{r}{2} = 9[/tex].
[tex]r = 18[/tex].
In other words, the radius of this circle is [tex]18\; {\rm cm}[/tex].
Substitute the value of [tex]r[/tex] back into either equation (for example, the equation for the arc length of this sector) to find [tex]\theta[/tex]:
[tex]\displaystyle \left(\frac{\theta}{2\, \pi}\right)\, 2\, \pi\, (18) = 12[/tex].
[tex]18\, \theta = 12[/tex].
[tex]\displaystyle \theta = \frac{12}{18} = \frac{2}{3}[/tex].
In other words, the arc of the sector in this question subtends an angle of [tex](2/3)[/tex] radians at the center of the circle.
Multiply the measure of the angle in radians by [tex](360^{\circ} / (2\, \pi))[/tex] to find the measure of this angle in degrees:
[tex]\begin{aligned}\theta = \frac{2}{3} \times \frac{360^{\circ}}{2\, \pi} \approx 38.2^{\circ}\end{aligned}[/tex].
