Question

gh is tangent to ⊙f. what is fh?
h
15 ft
g
8 ft
f
fh = ft
submit

Explanation:

Step1: Use the tangent - radius theorem

Since \( \overleftrightarrow{GH}\) is tangent to \( \odot F\) at \(G\), then \(FG\perp GH\) (a tangent to a circle is perpendicular to the radius at the point of tangency). So, \(\triangle FGH\) is a right - triangle with \(FG = 8\) ft (radius) and \(GH=15\) ft (tangent segment), and we want to find \(FH\) (hypotenuse).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right - triangle \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse and \(a\) and \(b\) are the legs of the right - triangle. In \(\triangle FGH\), let \(a = FG = 8\), \(b = GH = 15\), and \(c=FH\). Then \(FH^{2}=FG^{2}+GH^{2}\).
Substitute the values: \(FH^{2}=8^{2}+15^{2}\).
Calculate \(8^{2}=64\) and \(15^{2}=225\). So \(FH^{2}=64 + 225=289\).
Take the square root of both sides: \(FH=\sqrt{289}\).
Since \(17\times17 = 289\), \(FH = 17\) ft.

Answer:

\(17\)