Question

fh is tangent to circle g at point f. what is the length of the radius, r? 10 units 12 units 20 units 24 units

Explanation:

Step1: Recall tangent - radius property

Since $\overline{FH}$ is tangent to circle $G$ at point $F$, then $\angle{GFH}=90^{\circ}$ (tangent to a circle is perpendicular to the radius at the point of tangency), and triangle $GFH$ is a right - triangle.

Step2: Apply the Pythagorean theorem

In right - triangle $GFH$, by the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse and $a$ and $b$ are the legs. Here, $GH = 25 + r$, $FH = 35$, and $GF=r$. So, $r^{2}+35^{2}=(r + 25)^{2}$.

Step3: Expand the right - hand side

Expand $(r + 25)^{2}$ using the formula $(a + b)^{2}=a^{2}+2ab + b^{2}$. We get $r^{2}+35^{2}=r^{2}+50r+25^{2}$.

Step4: Simplify the equation

Subtract $r^{2}$ from both sides of the equation: $35^{2}=50r+25^{2}$. Calculate $35^{2}=1225$ and $25^{2}=625$. So, $1225=50r + 625$.

Step5: Solve for $r$

Subtract 625 from both sides: $1225−625 = 50r$, which gives $600 = 50r$. Then divide both sides by 50: $r=\frac{600}{50}=12$.

Answer:

12 units