which statements are true about circle q? select three options. the ratio of the measure of central angle…

Explanation:

Step1: Calculate ratio of central - angle

The measure of a full - circle is $360^{\circ}$. The measure of central angle $\angle PQR = 45^{\circ}$. The ratio of the measure of central angle $PQR$ to the measure of the entire circle is $\frac{45}{360}=\frac{1}{8}$.

Step2: Recall formula for area of a sector

The area of a sector of a circle with radius $r$ and central angle $\theta$ (in degrees) is $A=\frac{\theta}{360}\times\pi r^{2}$. So the area of the shaded sector depends on the length of the radius (since $r$ is in the formula) and also on the area of the circle ($A_{circle}=\pi r^{2}$ and $A_{sector}=\frac{\theta}{360}\times A_{circle}$).

Step3: Analyze incorrect statements

The area of the shaded sector with $\theta = 45^{\circ}$ and $r = 6$ is $A=\frac{45}{360}\times\pi\times6^{2}=\frac{1}{8}\times36\pi=\frac{9\pi}{2}\neq4$. Also, the ratio of the area of the shaded sector to the area of the circle is $\frac{\theta}{360}$, and the ratio of the length of the arc to the circumference of the circle is $\frac{\theta}{360}$, not the ratio of the length of the arc to the area of the circle.

Answer:

The ratio of the measure of central angle $PQR$ to the measure of the entire circle is $\frac{1}{8}$; The area of the shaded sector depends on the length of the radius; The area of the shaded sector depends on the area of the circle.