Question

the measure of central angle mnl is π radians, and the measure of the entire circle is 2π radians. the ratio of the measure of the central angle to the entire circle measure is dropdown. the area of the entire circle is dropdownπ units². the area of the sector is dropdownπ units². (circle with center n, radius 6, points m, l on the circle, sector mnl shaded)

Explanation:

Step1: Find the ratio of central angle to circle

The central angle is $\pi$ radians, the circle is $2\pi$ radians. The ratio is $\frac{\pi}{2\pi}=\frac{1}{2}$.

Step2: Calculate the area of the entire circle

The radius $r = 6$ (from the diagram, the length from N to L is 6, so radius is 6). The area of a circle is $A=\pi r^2$. So $A = \pi\times6^2 = 36\pi$ square units.

Step3: Calculate the area of the sector

The area of a sector is the ratio of the central angle to the full circle angle times the area of the circle. The ratio is $\frac{1}{2}$, so sector area is $\frac{1}{2}\times36\pi = 18\pi$ square units. But looking at the options (the dropdown has 1/2, etc.), for the ratio part, the first blank (ratio) is $\frac{1}{2}$, the area of the entire circle is $36\pi$ (but the second dropdown in the original problem might have been cut, but from the steps, the ratio is 1/2, the circle area is $36\pi$, and the sector area would be $\frac{1}{2}\times36\pi = 18\pi$, but if we consider the ratio for the sector area calculation, the ratio is 1/2.

Answer:

For the ratio of central angle to circle: $\frac{1}{2}$
For the area of the entire circle: $36\pi$ (assuming the second blank was about the circle's area formula result)
For the area of the sector (using the ratio): If we take the ratio $\frac{1}{2}$ and the circle area $36\pi$, the sector area is $18\pi$, but from the dropdown options (1/6, 1/2, 2), the ratio is $\frac{1}{2}$.