Question

analyzing circles given a central angle measure in degrees
in circle c, sq = 10 cm.
which statements about the circle are correct? check all that apply.
□ arc pq is congruent to arc sr.
□ the measure of arc qr is 150°.
□ the circumference of circle c is $20\pi$ cm.
□ arc ps measures about 13.1 cm.
□ arc qs measures about 15.7 cm.

Explanation:

Step1: Find circle radius

$SQ$ is a diameter, so radius $r = \frac{10}{2} = 5$ cm.

Step2: Verify circumference

Circumference formula: $C = 2\pi r$
$C = 2\pi \times 5 = 20\pi$ cm.

Step3: Analyze arc PQ and SR

No info shows $\angle PCQ = \angle SCR$, so arcs not congruent.

Step4: Calculate arc QR measure

$\angle PCQ = 30^\circ$, $\angle QCR$ is supplementary to $\angle PCS$ (no given congruence), so $\angle QCR
eq 150^\circ$.

Step5: Calculate arc PS length

Arc length formula: $L = \frac{\theta}{360^\circ} \times 2\pi r$. $\angle PCS = 180^\circ - 30^\circ = 150^\circ$
$L_{PS} = \frac{150^\circ}{360^\circ} \times 20\pi = \frac{25\pi}{3} \approx 26.18$ cm (not 13.1 cm).

Step6: Calculate arc QS length

QS is a semicircle, so $\theta = 180^\circ$
$L_{QS} = \frac{180^\circ}{360^\circ} \times 20\pi = 10\pi \approx 15.7$ cm.

Answer:

• The circumference of circle C is $20\pi$ cm.
• Arc QS measures about 15.7 cm.