Question

A circle has a center point upper C. Points upper D, upper E, upper F and upper G lie on the circumference of the circle.
If the radius of circle C
is 8 mm and m∠FCE=60°
, then find the exact measure of FG¯¯¯¯¯¯¯¯
.

(1 point)
FG=
mm

Answer 1

To find the length of \( \overline{FG} \) (the chord between points \( F \) and \( G \)), we can use the central angle \( \angle FCE = 60^\circ \).

Since \( F \) and \( G \) both lie on the circle centered at \( C \), we can apply the following formula for the length of the chord \( FG \):

\[
FG = 2 \cdot r \cdot \sin\left(\frac{\theta}{2}\right)
\]

where:
- \( r \) is the radius of the circle,
- \( \theta \) is the central angle (in this case \( \angle FCE \)).

Given:
- \( r = 8 \, \text{mm} \)
- \( \theta = 60^\circ \)

First, we calculate \( \frac{\theta}{2} \):

\[
\frac{60^\circ}{2} = 30^\circ
\]

Next, we substitute the values into the chord length formula:

\[
FG = 2 \cdot 8 \cdot \sin(30^\circ)
\]

We know that:

\[
\sin(30^\circ) = \frac{1}{2}
\]

Now substitute this value in:

\[
FG = 2 \cdot 8 \cdot \frac{1}{2}
\]
\[
FG = 2 \cdot 8 \cdot 0.5
\]
\[
FG = 8 \, \text{mm}
\]

Thus, the exact measure of \( \overline{FG} \) is:

\[
\boxed{8 \, \text{mm}}
\]