Question

in circle c, sq = 10 cm. which statements about the circle are correct? choose four correct answers. arc ps measures about 13.1 cm. arc pq is congruent to arc sr. the measure of arc qr is 150°. the circumference of circle c is 20π cm.

Explanation:

Step1: Find the radius

Given $SQ = 10$ cm, which is the diameter. So the radius $r=\frac{SQ}{2}=\frac{10}{2} = 5$ cm.

Step2: Calculate the circumference

The formula for the circumference of a circle is $C = 2\pi r$. Substituting $r = 5$ cm, we get $C=2\pi\times5=10\pi$ cm.

Step3: Calculate arc - length

The formula for the length of an arc is $l=\theta\times r$ (where $\theta$ is in radians). First, convert $30^{\circ}$ to radians: $30^{\circ}=\frac{\pi}{6}$ radians. If we assume we want to find the length of an arc with central - angle $\theta=\frac{\pi}{6}$ and $r = 5$ cm, then $l=\frac{\pi}{6}\times5=\frac{5\pi}{6}\approx2.62$ cm.

Step4: Analyze congruent arcs

If the central angles of two arcs in the same circle are equal, the arcs are congruent.

Step5: Find the measure of arc $QR$

The sum of angles around a point is $360^{\circ}$. If $\angle PCQ = 30^{\circ}$, and assume the circle is symmetric, and we know that the non - overlapping arcs make up the whole circle. If we assume the central angle of arc $QR$ is $\theta_{QR}$, and since the circle is symmetric and we have two $30^{\circ}$ central angles (e.g., $\angle PCQ$ and its opposite), then $\theta_{QR}=180^{\circ}- 30^{\circ}=150^{\circ}$.

Answer:

1. Arc $PQ$ is congruent to arc $SR$ (because they have equal central angles in the same circle).
2. The measure of arc $QR$ is $150^{\circ}$.
3. The circumference of circle $C$ is $10\pi$ cm (the statement in the option is incorrect as we calculated $C = 10\pi$ cm, not $20\pi$ cm).
4. Arc $PS$: First, the central angle of arc $PS$ is $120^{\circ}$. Convert $120^{\circ}$ to radians: $120^{\circ}=\frac{2\pi}{3}$ radians. With $r = 5$ cm, the arc - length $l=\frac{2\pi}{3}\times5=\frac{10\pi}{3}\approx10.47$ cm (the statement in the option is incorrect). So the correct answers are: Arc $PQ$ is congruent to arc $SR$, The measure of arc $QR$ is $150^{\circ}$.