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}{2pi}).
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 units².
the area of the sector is 100 units².
the area of the sector is more than half of the circle’s area.
( mangle rqp = 5 ) radians
radius = 10 units

Explanation:

Step1: Analyze central angle ratio

The measure of a full circle in radians is \(2\pi\). The central angle here is \(5\) radians. 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 sector area

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

Step3: Compare sector area to half - circle area

The area of a full circle is \(A_{circle}=\pi r^{2}=\pi\times10^{2}=100\pi\approx314.16\) square units. Half of the circle's area is \(\frac{1}{2}\times100\pi = 50\pi\approx157.08\) square units. Since \(250>157.08\), the sector area is more than half of the circle's area, 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 \(250\) units\(^{2}\).
• The area of the sector is more than half of the circle’s area.