Question

question 3
the circle shown has center o, circumference 144\pi, and diameters pr and qs. the length of arc ps is twice the length of arc pq. what is the length of arc qr?
a) 24\pi
b) 48\pi
c) 72\pi
d) 96\pi
note: figure not drawn to scale.

Explanation:

Step1: Recall arc - length relationship in a circle

The sum of the lengths of arcs in a circle is equal to the circumference of the circle. Let the length of arc $PQ=x$. Then the length of arc $PS = 2x$. Since $PR$ and $QS$ are diameters, arc $QR$ and arc $PS$ are equal (vertically - opposite central angles subtend equal arcs).

Step2: Set up an equation for the circumference

The sum of the lengths of arcs $PQ+PS + QR+RS$ is the circumference $C$ of the circle. Since $QR = PS$ and $RS=PQ$ (due to symmetry of diameters), $C=x + 2x+2x+x$. Given $C = 144\pi$.
So, $x + 2x+2x+x=144\pi$.
Combining like - terms, we get $6x = 144\pi$.

Step3: Solve for $x$

Dividing both sides of the equation $6x = 144\pi$ by 6, we have $x=\frac{144\pi}{6}=24\pi$.

Step4: Find the length of arc $QR$

Since $QR = 2x$, substituting $x = 24\pi$ into the equation, we get $QR=2\times24\pi = 48\pi$.

Answer:

B. $48\pi$