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Math  /  Geometry

Question9. In circle Q, m∠RQS=90∘Q, \mathrm{~m} \angle R Q S=90^{\circ} and the area of the shaded sector =9π=9 \pi. Find the length of QR‾\overline{Q R}.

Studdy Solution

STEP 1

What is this asking? We're given a circle with a shaded area, told the central angle of that area, and asked to find the circle's radius. Watch out! Remember the area of a sector formula depends on the angle being in radians, so we'll need to do a conversion!
Also, don't mix up radius and diameter!

STEP 2

1. Convert the angle to radians.
2. Set up the area equation.
3. Solve for the radius.

STEP 3

Alright, let's **convert** that angle to radians!
We're given 90∘90^\circ, and we know 360∘360^\circ is 2π2\pi radians.
So, to convert, we **multiply** 90∘90^\circ by 2π360∘\frac{2\pi}{360^\circ}.

STEP 4

90∘⋅2π360∘=180π360=π290^\circ \cdot \frac{2\pi}{360^\circ} = \frac{180\pi}{360} = \frac{\pi}{2} So, 90∘90^\circ is the same as π2\frac{\pi}{2} radians!

STEP 5

The **formula** for the area of a sector is 12⋅r2⋅θ\frac{1}{2} \cdot r^2 \cdot \theta, where rr is the **radius** and θ\theta is the **central angle** in radians.
We're given that the **shaded area** is 9π9\pi, and we just found that θ=π2\theta = \frac{\pi}{2}.
Let's **plug** those values into our formula!

STEP 6

9π=12⋅r2⋅π29\pi = \frac{1}{2} \cdot r^2 \cdot \frac{\pi}{2} Look at that beautiful equation!
Now, let's **solve** for rr!

STEP 7

First, let's **multiply** both sides of our equation by 22 to get rid of the 12\frac{1}{2}: 9π⋅2=12⋅r2⋅π2⋅29\pi \cdot 2 = \frac{1}{2} \cdot r^2 \cdot \frac{\pi}{2} \cdot 2 18π=r2⋅π218\pi = r^2 \cdot \frac{\pi}{2}

STEP 8

Now, let's **multiply** both sides by 2π\frac{2}{\pi} to isolate r2r^2: 18π⋅2π=r2⋅π2⋅2π18\pi \cdot \frac{2}{\pi} = r^2 \cdot \frac{\pi}{2} \cdot \frac{2}{\pi} 36=r236 = r^2

STEP 9

Finally, we take the **square root** of both sides to find rr: 36=r2\sqrt{36} = \sqrt{r^2} r=6r = 6Remember, the radius must be positive, so we only take the positive square root.

STEP 10

The length of QR‾\overline{QR}, which is the radius of the circle, is 6\textbf{6}.

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