Question

in a circle c, sq = 10 cm. arc pq is congruent to arc sr. the measure of arc qr is 150°. the circumference of circle c is 20π cm. arc qs measures about 15.7 cm. this question requires at least 4 answers.

Explanation:

Step1: Recall circle - arc properties

In a circle, if two arcs are subtended by equal central - angles, they are congruent. Since there is no information suggesting otherwise and assuming symmetry in the circle, arc $PQ$ and arc $SR$ are likely congruent.

Step2: Calculate arc measure

The central - angle of arc $QR$: The full - circle is $360^{\circ}$. If we assume the circle is symmetrically divided and one of the central - angles is $30^{\circ}$, and we want to find the measure of arc $QR$. Let's assume the circle is divided into four arcs by two intersecting lines at the center. If one central - angle is $30^{\circ}$, and we know that the sum of central - angles around a point is $360^{\circ}$. If we assume the circle is symmetrically divided, the measure of arc $QR$ cannot be $150^{\circ}$ without more information about the division of the circle.

Step3: Calculate circumference

Given $SQ = 10$ cm, if $SQ$ is the diameter $d$ of the circle, the circumference formula is $C=\pi d$. So $C = 10\pi\approx31.4$ cm, not $20\pi$ cm.

Step4: Calculate arc length

If $SQ$ is the diameter $d = 10$ cm, and arc $QS$ is a semi - arc. The formula for arc length $l$ of a semi - circle is $l=\frac{1}{2}\pi d$. Substituting $d = 10$ cm, we get $l=\frac{1}{2}\pi\times10 = 5\pi\approx15.7$ cm.

Answer:

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