Question

fh is tangent to circle g at point f. what is the length of the radius, r? 12 units 24 units 20 units 10 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 \(HG=r + 25\), leg \(FH = 35\) and leg \(FG=r\).

Step2: Apply Pythagorean theorem

In right - triangle FHG, by the Pythagorean theorem \(FH^{2}+FG^{2}=HG^{2}\). Substitute the values: \(35^{2}+r^{2}=(r + 25)^{2}\).

Step3: Expand the right - hand side

Expand \((r + 25)^{2}\) using the formula \((a + b)^{2}=a^{2}+2ab + b^{2}\), so \((r + 25)^{2}=r^{2}+50r+625\). The equation becomes \(35^{2}+r^{2}=r^{2}+50r+625\).

Step4: Simplify the equation

Since \(35^{2}=1225\), the equation \(1225+r^{2}=r^{2}+50r+625\). Subtract \(r^{2}\) from both sides of the equation. We get \(1225=50r + 625\).

Step5: Solve for r

Subtract 625 from both sides: \(1225−625=50r\), so \(600 = 50r\). Then divide both sides by 50: \(r=\frac{600}{50}=12\).

Answer:

12 units