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 the central angle for arc LM

The formula for the length of an arc is \( s = r\theta \) (where \( \theta \) is in radians) or \( s=\frac{\theta}{360^\circ}\times 2\pi r \) (where \( \theta \) is in degrees). Let's use the degree formula. For arc LM, \( s = 6.3\) cm, \( r = 6\) cm. Let the central angle for arc LM be \( \theta_{LM} \). So \( 6.3=\frac{\theta_{LM}}{360^\circ}\times 2\pi\times 6 \). We can also first find the central angle for arc MN (which is \( 75^\circ \)) and then find the total arc length of LMN as arc LM + arc MN.

First, let's find the length of arc MN. Using the arc length formula \( s=\frac{\theta}{360^\circ}\times 2\pi r \), where \( \theta = 75^\circ \), \( r = 6 \) cm.

Step2: Calculate length of arc MN

\( s_{MN}=\frac{75^\circ}{360^\circ}\times 2\pi\times 6 \)
Simplify: \( \frac{75}{360}\times 12\pi=\frac{75\times 12\pi}{360}=\frac{900\pi}{360}=2.5\pi\approx 2.5\times 3.1416 = 7.854 \) cm. Wait, but arc LM is 6.3 cm. Then total arc LMN is \( 6.3 + 7.854\approx 14.154 \approx 14.2 \) cm? Wait, no, maybe I made a mistake. Wait, the central angle for arc LM: let's calculate it. Using \( s = \frac{\theta}{360}\times 2\pi r \), so \( \theta=\frac{s\times 360}{2\pi r} \). For arc LM, \( s = 6.3 \), \( r = 6 \). So \( \theta_{LM}=\frac{6.3\times 360}{2\pi\times 6}=\frac{6.3\times 60}{\pi}=\frac{378}{\pi}\approx 120.3^\circ \) (since \( \pi\approx 3.1416 \), \( 378\div 3.1416\approx 120.3 \)). Then the total central angle for arc LMN is \( \theta_{LM}+\theta_{MN}\approx 120.3^\circ + 75^\circ = 195.3^\circ \). Then arc length \( s=\frac{195.3}{360}\times 2\pi\times 6 \). Let's calculate that: \( \frac{195.3}{360}\times 12\pi=\frac{195.3\times 12\pi}{360}=\frac{195.3\times \pi}{30}\approx \frac{195.3\times 3.1416}{30}\approx \frac{613.6}{30}\approx 20.45 \)? No, that's not matching. Wait, maybe I messed up the diagram. Wait, the diagram shows O, L, M, N with angle MON is 75 degrees. So arc LMN is from L to M to N, so arc LM + arc MN. Wait, maybe the first approach: arc MN length. Let's recalculate arc MN: \( s = \frac{75}{360}\times 2\pi\times 6=\frac{75\times 12\pi}{360}=\frac{900\pi}{360}=2.5\pi\approx 7.85 \) cm. Arc LM is 6.3 cm. So total arc LMN is 6.3 + 7.85 = 14.15 ≈14.2 cm. So the answer should be 14.2 cm.

Answer:

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