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 circles area.
$mangle rqp = 5$ radians
radius = 10 units

Explanation:

Step1: Find central angle ratio

The total measure of a circle is $2\pi$ radians. The central angle is 5 radians, so the ratio is $\frac{5}{2\pi}$.

Step2: Calculate sector area

Use sector area formula $A=\frac{1}{2}r^2\theta$, where $r=10$, $\theta=5$.
$A=\frac{1}{2}\times10^2\times5=\frac{1}{2}\times100\times5=250$ units$^2$

Step3: Compare to half the circle area

Half the circle area is $\frac{1}{2}\pi r^2=\frac{1}{2}\pi\times10^2=50\pi\approx157.08$ units$^2$. Since $250>157.08$, the sector area is more than half the circle's area.

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.