Question

in circle o, the length of radius ol is 6 cm and the length of arc lm is 6.3 cm. the measure of angle mon is 75°. rounded to the nearest tenth of a centimeter, what is the length of arc lmn?
○ 7.9 cm
○ 10.2 cm
○ 12.6 cm
○ 14.2 cm

Explanation:

Step1: Find arc MN length

Arc length formula: $s = r\theta$ (θ in radians). Convert $75^\circ$ to radians: $75\times\frac{\pi}{180}=\frac{5\pi}{12}$. Radius $r = 6$. So arc MN: $6\times\frac{5\pi}{12}=\frac{5\pi}{2}\approx7.9$ cm.

Step2: Find arc LMN length

Arc LMN = arc LM + arc MN. Arc LM = 6.3 cm, arc MN ≈7.9 cm. So 6.3 + 7.9 = 14.2? Wait, no, wait. Wait, maybe I messed up. Wait, no, let's recalculate arc MN. Wait, $75^\circ$ in radians: $75\times\frac{\pi}{180}=\frac{5\pi}{12}\approx1.308997$. Then arc MN: $6\times1.308997\approx7.854$ cm. Then arc LMN is arc LM (6.3) + arc MN (≈7.854) = 14.154 ≈14.2 cm? Wait, but let's check again. Wait, maybe the angle for arc LM? Wait, no, the problem says arc LM is 6.3 cm, angle MON is 75 degrees. So arc LMN is from L to M to N, so arc LM (6.3) plus arc MN. Arc MN: central angle 75 degrees, radius 6. So arc length formula is $s=\frac{\theta}{360}\times2\pi r$. So for arc MN: $\frac{75}{360}\times2\pi\times6=\frac{75\times12\pi}{360}=\frac{900\pi}{360}=2.5\pi\approx7.85$ cm. Then arc LMN: 6.3 + 7.85 ≈14.15 ≈14.2 cm.

Answer:

14.2 cm (corresponding to the option "14.2 cm")