Question

which statements are true? check all that apply.
the ratio of the measure of the central angle to the measure of the entire circle is \\(\frac{5}{2\pi}\\).
the ratio of the measure of the central angle to the measure of the entire circle is \\(\frac{5}{2}\\).
the area of the sector is 250 units\\(^2\\).
the area of the sector is 100 units\\(^2\\).
the area of the sector is more than half of the circle’s area.
\\(m\angle rqp = 5\\) radians
radius = 10 units

Explanation:

Step1: Analyze the central angle ratio

The measure of a full circle in radians is \(2\pi\). The central angle here is \(5\) radians. So the ratio of the central angle to the full circle is \(\frac{5}{2\pi}\). So the first statement is true, the second is false.

Step2: Calculate the area of the sector

The formula for the area of a sector when the central angle is in radians is \(A=\frac{1}{2}r^{2}\theta\), where \(r = 10\) and \(\theta=5\). Plugging in the values: \(A=\frac{1}{2}\times10^{2}\times5=\frac{1}{2}\times100\times5 = 250\) square units. So the third statement (area is 250) is true, the fourth (100) is false.

Step3: Compare sector area to half the circle's area

The area of the full circle is \(A_{circle}=\pi r^{2}=\pi\times10^{2} = 100\pi\approx314.16\) square units. Half of the circle's area is \(\frac{100\pi}{2}=50\pi\approx157.08\) square units. The sector area is 250, which is more than \(50\pi\approx157.08\). So the fifth statement is true.

Answer:

• The ratio of the measure of the central angle to the measure of the entire circle is \(\frac{5}{2\pi}\).
• The area of the sector is 250 units².
• The area of the sector is more than half of the circle’s area.