Question

which statements are true about circle q? select three options. 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 is 4 units\\(^2\\) 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. the ratio of the area of the shaded sector to the area of the circle is equal to the ratio of the length of the arc to the area of the circle.

Explanation:

Step1: Analyze the central angle ratio

A full circle is \(360^\circ\). The central angle \(\angle PQR = 45^\circ\). The ratio is \(\frac{45}{360}=\frac{1}{8}\). So this statement is true.

Step2: Calculate the area of the shaded sector

The formula for the area of a sector is \(A=\frac{\theta}{360^\circ}\times\pi r^2\), where \(\theta = 45^\circ\) and \(r = 6\)? Wait, no, wait the radius here is 6? Wait no, wait the diagram shows radius 6? Wait no, wait the problem: wait the shaded sector has central angle \(45^\circ\), radius \(r = 6\)? Wait no, wait the first option: ratio is \(\frac{45}{360}=\frac{1}{8}\), that's correct. Now the area of the shaded sector: \(A=\frac{45}{360}\times\pi\times6^2=\frac{1}{8}\times36\pi = 4.5\pi\approx14.14\), so the statement "The area of the shaded sector is 4 units²" is false.

Step3: Analyze the dependence on radius

The area of a sector is \(A=\frac{\theta}{360^\circ}\times\pi r^2\), so it depends on the radius. So "The area of the shaded sector depends on the length of the radius" is true.

Step4: Analyze dependence on circle's area

The area of the sector is a fraction of the circle's area (\(\frac{\theta}{360^\circ}\) times the circle's area). So if the circle's area changes (due to radius change), the sector's area changes. So "The area of the shaded sector depends on the area of the circle" is true.

Step5: Analyze the arc length to circle area ratio

The ratio of the area of the sector to the circle's area is \(\frac{\theta}{360^\circ}\), and the ratio of the arc length to the circumference is also \(\frac{\theta}{360^\circ}\), but the arc length is a length and the circle's area is an area, so their ratio is not equal to the sector - circle area ratio. So this statement is false.

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