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

Answer:

12 units

Explanation:

Step1: Recall tangent - radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. So, $\triangle FHG$ is a right - triangle with hypotenuse $HG = r + 25$ and legs $FH=35$ and $FG = r$.

Step2: Apply Pythagorean theorem

By the Pythagorean theorem, $FH^{2}+FG^{2}=HG^{2}$. Substituting the values, we get $35^{2}+r^{2}=(r + 25)^{2}$.

Step3: Expand the equation

Expand $(r + 25)^{2}$: $35^{2}+r^{2}=r^{2}+50r + 25^{2}$.

Step4: Simplify the equation

$1225+r^{2}=r^{2}+50r + 625$. Subtract $r^{2}$ from both sides: $1225=50r + 625$.

Step5: Solve for r

Subtract 625 from both sides: $1225−625 = 50r$, so $600 = 50r$. Then $r=\frac{600}{50}=12$ units.