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$In circle Q,QR=12 and m/_RQS=20^(@). Find the length of RS. Express your answer as a fraction times pi.$

In circle $Q, Q R=12$ and $\mathrm{m} \angle R Q S=20^{\circ}$ . Find the length of $R S$ . Express your answer as a fraction times $\pi$ .

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Find the roots of factored polynomials

Full solution

Q. In circle

Q, Q R=12

and

\mathrm{m} \angle R Q S=20^{\circ}

. Find the length of

R S

. Express your answer as a fraction times

\pi

.

Calculate Arc Length Formula: To find RS, we need to use the formula for the arc length, which is $(\theta/360) \times 2\pi r$ , where $\theta$ is the central angle in degrees and $r$ is the radius of the circle.
Determine Radius: Since $QR$ is the radius and $QR=12$ , we have $r=12$ .
Find Central Angle: The central angle $\theta$ is twice the given angle $m/\_RQS$ because the angle at the center of a circle is twice any angle at the circumference standing on the same arc. So, $\theta=2\times m/\_RQS=2\times 20=40$ degrees.
Calculate Arc Length RS: Now we can calculate the arc length RS: $RS = (\theta/360) \times 2\pi r = (40/360) \times 2\pi\times12$ .
Simplify Fraction: Simplify the fraction $\frac{40}{360}$ to $\frac{1}{9}$ .
Multiply by $2\pi r$ : Now multiply $(1/9)$ by $2\pi\cdot 12$ to get RS: $RS = (1/9) \cdot 2\pi\cdot 12 = (2/9) \cdot \pi \cdot 12$ .
Final Calculation: Finally, multiply $(2/9)$ by $12$ to get RS: $RS = (2/9) \times \pi \times 12 = (24/9) \times \pi = (8/3) \times \pi$ .

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