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Which statements are true? Check all that apply.

A. The ratio of the measure of the central angle to the measure of the entire circle is [tex]$\frac{5}{2 \pi}$[/tex].
B. The ratio of the measure of the central angle to the measure of the entire circle is [tex]$\frac{5}{2}$[/tex].
C. The area of the sector is 250 units [tex]${ }^2$[/tex].
D. The area of the sector is 100 units.
E. The area of the sector is more than half of the circle's area.
F. Divide the measure of the central angle by [tex]$2 \pi$[/tex] to find the ratio.

Sagot :

GinnyAnswer
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.
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