Question

what is fi?
fi = \square

Explanation:

Step1: Use the Angle - Angle - Side (AAS) congruence theorem

Since \(FH\) is the angle - bisector (\(\angle IFH=\angle GFH\)), \(\angle FIH = \angle FGH=90^{\circ}\), and \(FH = FH\) (common side), \(\triangle FIH\cong\triangle FGH\) (AAS). Then \(FI = FG\).

Step2: Set up the equation

Given \(FI=2v + 24\) and \(FG = 10v\). Because \(FI = FG\), we have the equation \(2v+24=10v\).

Step3: Solve the equation for \(v\)

Subtract \(2v\) from both sides: \(24=10v - 2v\), so \(24 = 8v\). Then \(v=\frac{24}{8}=3\).

Step4: Calculate \(FI\)

Substitute \(v = 3\) into \(FI=2v + 24\). \(FI=2\times3+24=6 + 24=30\).

Answer:

\(30\)