Question

in the accompanying diagram, \\(\overline{afb}\\), \\(\overline{aec}\\), and \\(\overline{bgc}\\) are tangent to circle \\(o\\) at \\(f\\), \\(e\\), and \\(g\\), respectively. if \\(ab = 32\\), \\(ae = 20\\), and \\(ec = 24\\), find \\(bc\\).

Explanation:

Step1: Recall Tangent Segment Theorem

Tangent segments from a common external point to a circle are congruent. So, \( AE = AF = 20 \), \( EC = GC = 24 \), and \( BF = BG \).

Step2: Find \( BF \)

Given \( AB = 32 \) and \( AF = 20 \), we can find \( BF \) by subtracting \( AF \) from \( AB \).
\( BF = AB - AF = 32 - 20 = 12 \).

Step3: Find \( BC \)

Since \( BG = BF = 12 \) and \( GC = 24 \), we find \( BC \) by adding \( BG \) and \( GC \).
\( BC = BG + GC = 12 + 24 = 36 \).

Answer:

\( 36 \)