Question

explore the relationship between the radius and area of a circle by following these steps. 1. press continue to rearrange the sectors of the circle to form a shape resembling a parallelogram. 2. if we divide the circle into more sectors, the bottom shape more closely resembles a parallelogram. press continue to repeat the last step with 16 sectors. 3. the formula for the area of a parallelogram is base × height. 4. the height of the parallelogram is dropdown with options 2, πr, r

Explanation:

Step1: Recall the circle - parallelogram transformation

When we divide a circle into sectors and rearrange them to form a parallelogram, the height of the parallelogram is equal to the radius of the original circle. This is because as we increase the number of sectors, the "height" of the parallelogram - like shape formed from the circle's sectors corresponds to the radius of the circle. The base of the parallelogram will be related to the circumference of the circle (specifically half of the circumference, i.e., $\pi r$), but for the height, since we are stacking the sectors in a way that the vertical distance from the base to the top of the parallelogram is the same as the radius of the circle.

Step2: Analyze the options

The options are 2, $\pi r$, and $r$. We know from the transformation of the circle into a parallelogram that the height of the parallelogram is equal to the radius $r$ of the circle. The value 2 is not related to the circle's properties in this context, and $\pi r$ is related to the base (half of the circumference $C = 2\pi r$, so half of it is $\pi r$) rather than the height.

Answer:

The height of the parallelogram is $r$. So the correct option (from the dropdown) is $r$.