Question

points a, b and c lie on a circle with center q.

• the area of sector aqb is twice the area of sector bqc.
• the length of arc ab is 28 centimeters.

what is the length, in centimeters, of arc bc?
centimeters

Explanation:

Step1: Recall sector area and arc length formulas

The area of a sector with central angle \(\theta\) (in radians) and radius \(r\) is \(A = \frac{1}{2}r^{2}\theta\), and the arc length \(s\) is \(s = r\theta\). For two sectors with the same radius (since they are in the same circle with center \(Q\)), the ratio of their areas is equal to the ratio of their central angles, and the ratio of their arc lengths is also equal to the ratio of their central angles.

Let the area of sector \(BQC\) be \(A_1\) with central angle \(\theta_1\), and the area of sector \(AQB\) be \(A_2\) with central angle \(\theta_2\). We know \(A_2 = 2A_1\). Using the sector area formula \(A=\frac{1}{2}r^{2}\theta\), for the same radius \(r\), \(\frac{A_2}{A_1}=\frac{\frac{1}{2}r^{2}\theta_2}{\frac{1}{2}r^{2}\theta_1}=\frac{\theta_2}{\theta_1}\). So \(\frac{\theta_2}{\theta_1}=\frac{2A_1}{A_1} = 2\), which means \(\theta_2 = 2\theta_1\).

Step2: Relate arc lengths to central angles

The arc length \(s = r\theta\). Let the arc length of \(BC\) be \(s_1=r\theta_1\) and the arc length of \(AB\) be \(s_2 = r\theta_2\). Substituting \(\theta_2 = 2\theta_1\) into the arc length formula for \(AB\), we get \(s_2=r(2\theta_1)=2(r\theta_1)=2s_1\).

Step3: Solve for arc length of \(BC\)

We know \(s_2 = 28\) cm (arc length of \(AB\)) and \(s_2=2s_1\). So we can solve for \(s_1\) by rearranging the equation: \(s_1=\frac{s_2}{2}\). Substituting \(s_2 = 28\) into this, we get \(s_1=\frac{28}{2}=14\) cm.

Answer:

14