point f is on circle c. what is the length of line segment gf? 12.5 units 15.0 units 17.5 units 20.0 units

Answer:

17.5 units

Explanation:

Step1: Recall tangent - radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. So, $\angle GEC = 90^{\circ}$.

Step2: Use the Pythagorean theorem in right - triangle $GEC$

In right - triangle $GEC$, $GC^{2}=GE^{2}+EC^{2}$. Given $GE = 10$ and $EC=7.5$. Then $GC=\sqrt{10^{2}+7.5^{2}}=\sqrt{100 + 56.25}=\sqrt{156.25}=12.5$.

Step3: Find the length of $GF$

Since $GF=GC + CF$ and $CF$ is the radius of the circle with $CF = 7.5$ and $GC = 12.5$. So, $GF=12.5+7.5 = 17.5$ units.