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: Analyze SQ as diameter

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

Step2: Check Arc PQ and Arc SR

\( \angle PCQ = 30^\circ \), vertical angles are equal, so \( \angle SCR=\angle PCQ = 30^\circ \). Arcs with equal central angles are congruent. Arc \( PQ \) and arc \( SR \) have central angles \( 30^\circ \), so they are congruent.

Step3: Measure of arc QR

Central angle for arc \( QR \): \( 180^\circ - 30^\circ=150^\circ \), so measure of arc \( QR \) is \( 150^\circ \).

Step4: Circumference of circle C

Circumference formula \( C = \pi d \), \( d = 10\space\text{cm} \), so \( C = 10\pi\space\text{cm} \), not \( 20\pi \). So this statement is wrong.

Step5: Arc PS length

Central angle for arc \( PS \): \( 180^\circ - 30^\circ = 150^\circ \). Arc length formula \( L=\frac{\theta}{360^\circ}\times 2\pi r \), \( \theta = 150^\circ \), \( r = 5\space\text{cm} \). \( L=\frac{150^\circ}{360^\circ}\times 2\pi\times 5=\frac{5}{12}\times 10\pi=\frac{25\pi}{6}\approx 13.1\space\text{cm} \).

Step6: Arc QS length

Arc \( QS \) is a semicircle? No, \( QS \) is diameter, arc \( QS \) is a semicircle? Wait, \( QS \) is diameter, so arc \( QS \) length is \( \frac{1}{2}\times 2\pi r=\pi r = 5\pi\approx 15.7\space\text{cm} \)? Wait, no: \( QS \) is diameter, so the arc \( QS \) (semicircle) length is \( \pi r=5\pi\approx 15.7\space\text{cm} \). Wait, but let's recalculate: arc length for semicircle is \( \pi d/2=\pi\times 10/2 = 5\pi\approx 15.7\space\text{cm} \). But wait, earlier we thought \( SQ \) is diameter, so arc \( QS \) is semicircle, length \( 5\pi\approx 15.7\space\text{cm} \). But wait, let's check each statement again.

Wait, step4: circumference is \( \pi d = 10\pi \), so the statement "circumference is \( 20\pi \)" is wrong.

Now, let's list correct statements:

- Arc PQ is congruent to arc SR: correct (equal central angles).
- The measure of arc QR is \( 150^\circ \): correct (central angle \( 180 - 30 = 150 \)).
- Arc PS measures about \( 13.1\space\text{cm} \): correct (calculated as \( \frac{150}{360}\times 2\pi\times 5\approx 13.1 \)).
- Arc QS measures about \( 15.7\space\text{cm} \): correct (semicircle, length \( 5\pi\approx 15.7 \)). Wait, but earlier I thought \( QS \) is diameter, so arc \( QS \) is semicircle, length \( 5\pi\approx 15.7 \). But let's check the initial problem: \( SQ = 10\space\text{cm} \) (diameter), so arc \( QS \) is a semicircle, length \( \pi\times 10/2 = 5\pi\approx 15.7 \). So this is correct.

Wait, but let's re-express each:

1. Arc PQ ≅ arc SR: correct (central angles 30°).
2. Arc QR: 150°: correct (180 - 30 = 150).
3. Circumference: 10π, not 20π: wrong.
4. Arc PS: ~13.1 cm: correct.
5. Arc QS: ~15.7 cm: correct (semicircle, 5π≈15.7).

Wait, but let's check arc QS: \( QS \) is diameter, so the arc \( QS \) (the semicircle) has length \( \pi r = 5\pi\approx 15.7 \). Yes.

So correct statements: Arc PQ is congruent to arc SR, The measure of arc QR is \( 150^\circ \), Arc PS measures about 13.1 cm, Arc QS measures about 15.7 cm. Wait, but let's check again:

Wait, arc QS: \( QS \) is diameter, so the arc from Q to S through the bottom (or top) is a semicircle. So length is \( \pi d/2 = 5\pi\approx 15.7 \). Correct.

Arc PS: central angle 150°, length \( \frac{150}{360}\times 2\pi\times 5=\frac{5}{12}\times 10\pi=\frac{25\pi}{6}\approx 13.1 \). Correct.

Arc PQ and arc SR: central angles 30°, so congruent. Correct.

Arc QR: central angle 150°, correct.

Circumference: \( \pi d = 10\pi \), so the statement "20π" is wrong.

So correct options:

- Arc PQ is congruent to arc SR.
- The measure of arc QR is \( 150^\circ \).
- Arc PS measures about 13.1 cm.
- Arc QS measures about 15.7 cm.

Wait, but let's confirm each:

1. Arc PQ ≅ arc SR: Central angles ∠PCQ and ∠SCR are vertical angles, so equal (30°), so arcs are congruent. Correct.

2. Measure of arc QR: ∠QCR = 180° - 30° = 150°, so arc QR is 150°. Correct.

3. Circumference: \( C = \pi d = 10\pi \), so "20π" is wrong.

4. Arc PS: central angle 150°, length \( \frac{150}{360} \times 2\pi \times 5 = \frac{25\pi}{6} \approx 13.1 \). Correct.

5. Arc QS: QS is diameter, so arc QS is a semicircle? Wait, QS is diameter, so the arc from Q to S (the semicircle) has length \( \pi r = 5\pi \approx 15.7 \). Correct.

So the correct statements are:

- Arc PQ is congruent to arc SR.
- The measure of arc QR is \( 150^\circ \).
- Arc PS measures about 13.1 cm.
- Arc QS measures about 15.7 cm.

Answer:

• Arc PQ is congruent to arc SR.
• The measure of arc QR is \( 150^\circ \).
• Arc PS measures about 13.1 cm.
• Arc QS measures about 15.7 cm.