त्ेㄱ $\newline$ $\odot O$ is divided into three sectors. Points $P, Q$ , and $R$ are on the circle. Sector $P O R$ has an area of $4 \pi$ and sector $R O Q$ has an area of $3 \pi$ . If the radius of the circle is $3$ , what is the measure of the central angle of minor sector $Q O P$ , in degrees?

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Q. त्ेㄱ

\newline

\odot O

is divided into three sectors. Points

P, Q

, and

R

are on the circle. Sector

P O R

has an area of

4 \pi

and sector

R O Q

has an area of

3 \pi

. If the radius of the circle is

3

, what is the measure of the central angle of minor sector

Q O P

, in degrees?

Understand Proportional Area Formula: First, we need to understand that the area of a sector of a circle is proportional to the central angle that subtends the sector. The formula for the area of a sector with central angle $\theta$ (in radians) and radius $r$ is given by $A = (\theta/2\pi) * \pi r^2$ . Since we are given the areas of two sectors and the radius of the circle, we can use this information to find the central angles of these sectors.
Calculate Central Angle for Sector POR: The area of sector POR is given as $4\pi$ . Using the formula $A = (\theta/2\pi) * \pi r^2$ , we can substitute the given values to find the central angle for sector POR. Let's denote the central angle for sector POR as $\theta_{POR}$ . We have $4\pi = (\theta_{POR}/2\pi) * \pi * 3^2$ . Simplifying this, we get $\theta_{POR} = (4\pi * 2\pi) / (\pi * 9)$ . The $\pi$ 's cancel out, and we are left with $\theta_{POR} = (8/9) * 2\pi$ .
Calculate Central Angle for Sector ROQ: Similarly, the area of sector ROQ is given as $3\pi$ . Let's denote the central angle for sector ROQ as $\theta_{\text{ROQ}}$ . Using the same formula, we have $3\pi = (\theta_{\text{ROQ}}/2\pi) * \pi * 3^2$ . Simplifying this, we get $\theta_{\text{ROQ}} = (3\pi * 2\pi) / (\pi * 9)$ . The $\pi$ 's cancel out, and we are left with $\theta_{\text{ROQ}} = (6/9) * 2\pi$ .
Find Central Angle for Sector QOP: Now, we need to find the central angle for the minor sector QOP. The total angle in a circle is $2\pi$ radians, which is equivalent to $360$ degrees. Since we have the central angles for sectors POR and ROQ, we can subtract these from the total angle to find the central angle for sector QOP. Let's denote the central angle for sector QOP as $\theta_{\text{QOP}}$ . We have $\theta_{\text{QOP}} = 2\pi - (\theta_{\text{POR}} + \theta_{\text{ROQ}})$ .
Convert Radians to Degrees: Substituting the values we found for $\theta_{POR}$ and $\theta_{ROQ}$ , we get $\theta_{QOP} = 2\pi - ((\frac{8}{9}) * 2\pi + (\frac{6}{9}) * 2\pi)$ . Simplifying this, we get $\theta_{QOP} = 2\pi - (\frac{14}{9}) * 2\pi$ . This simplifies to $\theta_{QOP} = (2\pi * (\frac{9}{9})) - (\frac{14}{9}) * 2\pi$ , which further simplifies to $\theta_{QOP} = (\frac{$18$}{$9$}) * $2$\pi - (\frac{$14$}{$9$}) * $2$\pi.
Calculate Final Central Angle: Continuing the simplification, we get $\theta_{QOP} = \frac{4}{9} \times 2\pi$. To convert this angle from radians to degrees, we multiply by $\left(\frac{180}{\pi}\right)$. So, $\theta_{QOP}$ in degrees is $\frac{4}{9} \times 2\pi \times \left(\frac{180}{\pi}\right)$. The $\pi$'s cancel out, and we are left with $\theta_{QOP} = \frac{4}{9} \times 2 \times 180$.
Calculate Final Central Angle: Continuing the simplification, we get $\theta_{QOP} = \frac{4}{9} \times 2\pi$. To convert this angle from radians to degrees, we multiply by $\left(\frac{180}{\pi}\right)$. So, $\theta_{QOP}$ in degrees is $\left(\frac{4}{9} \times 2\pi \times \frac{180}{\pi}\right)$. The $\pi$'s cancel out, and we are left with $\theta_{QOP} = \left(\frac{4}{9} \times 2 \times 180\right)$. Finally, we calculate $\theta_{QOP} = \left(\frac{4}{9} \times 360\right)$. Multiplying this out, we get $\theta_{QOP} = 4 \times 40$, which equals $160$ degrees. Therefore, the measure of the central angle of minor sector QOP is $160$ degrees.