In triangle $FGH$, an angle bisector from vertex $G$ is perpendicular to side $FH$. If $FG=8$ and $GH=16$, what is the length of $FH$?

• A. $8$
• B. $16$
• C. $4$
• D. $24$

Answer: (B)

Verified step by step guidance

1. Identify the given elements: In triangle FGH, the angle bisector from vertex G is perpendicular to side FH. The lengths FG = 8 and GH = 16 are given, and we need to find the length of FH.
2. Recall the Angle Bisector Theorem, which states that the angle bisector divides the opposite side into segments proportional to the adjacent sides. If the angle bisector from G meets FH at point I, then \( \frac{FI}{IH} = \frac{FG}{GH} = \frac{8}{16} = \frac{1}{2} \).
3. Since the angle bisector is perpendicular to FH, triangle GIH is a right triangle with GI as the angle bisector and GI \perp FH. This means GI is the height from G to FH.
4. Let the length of FH be \( x \). Using the ratio from the angle bisector theorem, we can express the segments as \( FI = \frac{x}{3} \) and \( IH = \frac{2x}{3} \).
5. Apply the Pythagorean theorem to triangles FGI and GHI, using the known lengths FG = 8 and GH = 16, and the segments FI and IH expressed in terms of \( x \). Set up equations and solve for \( x \), the length of FH.