To find the measure of arc EC, we can use the relationship between the angles and arcs in a circle.
The measure of angle EGC is given as:
\[ \text{Measure of angle EGC} = \frac{1}{2}(\text{Measure of arc EC} - \text{Measure of arc DF}) \]
We know:
- Measure of arc EC = \(5x\)
- Measure of angle EGC = \(7x\)
- Measure of arc DF = \(90^\circ\)
Substituting these values into our equation, we have:
\[ 7x = \frac{1}{2}(5x - 90) \]
Next, we solve for \(x\). First, multiply both sides by 2 to eliminate the fraction:
\[ 14x = 5x - 90 \]
Now, isolate \(x\) by subtracting \(5x\) from both sides:
\[ 14x - 5x = -90 \] \[ 9x = -90 \] \[ x = -10 \]
Now we can find the measure of arc EC:
\[ \text{Measure of arc EC} = 5x = 5(-10) = -50 \]
This result indicates we made a mistake since an arc cannot have a negative measure.
Let’s re-examine the equation.
The correct relationship based on the formula can also be seen as:
\[ 7x = \frac{1}{2}(\text{arc EC} - \text{arc DF}) \] However, we should always ensure that we substitute correctly, so let’s use:
\[ 7x = \frac{1}{2}(5x + 90) \]
This leads to:
\[ 14x = 5x + 90 \] \[ 14x - 5x = 90 \] \[ 9x = 90 \] \[ x = 10 \]
Now substituting back to find arc EC:
\[ \text{Measure of arc EC} = 5x = 5(10) = 50 \]
So the measure of arc EC is:
\[ \text{Measure of arc EC} = 50^\circ \]
Thus, the correct answer is:
\[ \boxed{50^\circ} \]
2 answers