Let's evaluate the given statements one by one using the provided true numerical results.
### Statement 1:
"The ratio of the measure of the central angle to the measure of the entire circle is [tex]\(\frac{5}{2 \pi}\)[/tex]."
The true ratio of the central angle (5 radians) to the entire circle (which is [tex]\(2\pi\)[/tex] radians) is indeed [tex]\(\frac{5}{2 \pi}\)[/tex].
The provided result confirms that the ratio is approximately 0.7957747154594768. Therefore, this statement is true.
### Statement 2:
"The ratio of the measure of the central angle to the measure of the entire circle is [tex]\(\frac{5}{2}\)[/tex]."
The calculation of [tex]\(\frac{5}{2 \pi}\)[/tex] results in approximately 0.7957747154594768. However, [tex]\(\frac{5}{2}\)[/tex] equals 2.5.
Since these two values differ significantly, this statement is false.
### Statement 3:
"The area of the sector is 250 units [tex]\({}^2\)[/tex]."
The sector area given is 250 units [tex]\({}^2\)[/tex], aligning perfectly with the true area calculated.
Therefore, this statement is true.
### Statement 4:
"The area of the sector is 100 units."
The area of the sector is given as 250 units [tex]\({}^2\)[/tex], not 100 units [tex]\({}^2\)[/tex].
Thus, this statement is false.
### Statement 5:
"The area of the sector is more than half of the circle's area."
The true calculations confirm that 250 units [tex]\({}^2\)[/tex] (the area of the sector) is indeed more than half of the circle's area. The check `is_more_than_half` is marked as true.
Therefore, this statement is true.
### Statement 6:
"Divide the measure of the central angle by [tex]\(2\pi\)[/tex] to find the ratio."
To find the ratio of the central angle to the full circle, you indeed need to divide the central angle [tex]\(5\)[/tex] by [tex]\(2\pi\)[/tex].
Thus, this statement is true.
### Summary:
True statements are:
- The ratio of the measure of the central angle to the measure of the entire circle is [tex]\(\frac{5}{2 \pi}\)[/tex].
- The area of the sector is 250 units [tex]\({}^2\)[/tex].
- The area of the sector is more than half of the circle's area.
- Divide the measure of the central angle by [tex]\(2 \pi\)[/tex] to find the ratio.
