Question

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 20π cm.
arc ps measures about 13.1 cm.
arc qs measures about 15.7 cm.

Explanation:

Step1: Determine the radius

Given \(SQ = 10\) cm (diameter), so radius \(r=\frac{SQ}{2}=5\) cm.

Step2: Check arc congruence

\(\angle PCQ=\angle RCS = 30^{\circ}\), so arc \(PQ\) is congruent to arc \(SR\) (equal central angles in a circle have congruent arcs).

Step3: Calculate arc \(QR\) measure

\(\angle QCR = 180^{\circ}- 30^{\circ}=150^{\circ}
eq150^{\circ}\) (incorrect).

Step4: Calculate circumference

Circumference \(C = 2\pi r=2\pi\times5 = 10\pi
eq20\pi\) (incorrect).

Step5: Calculate arc \(PS\) length

\(\angle PCS=180^{\circ}-30^{\circ}=150^{\circ}\), arc length \(l=\frac{\theta}{360}\times2\pi r=\frac{150}{360}\times2\pi\times5=\frac{125.66}{3}\approx13.1\) cm.

Step6: Calculate arc \(QS\) length

\(\angle QCS = 90^{\circ}\), arc length \(l=\frac{90}{360}\times2\pi\times5=\frac{31.4}{2}=15.7\) cm.

Answer:

Arc PQ is congruent to arc SR.
Arc PS measures about 13.1 cm.
Arc QS measures about 15.7 cm.