Question

if \\( \angle f h g \cong \angle f h i \\), \\( f g = 8 u - 75 \\), and \\( f i = 2 u - 15 \\), what is \\( f g \\)?

Explanation:

Step1: Use the Angle - Bisector Theorem (distance from a point to sides of an angle)

Since \(\angle FHG\cong\angle FHI\), and \(FG\perp HG\), \(FI\perp HI\), then \(FG = FI\) (a point on the bisector of an angle is equidistant from the sides of the angle).
So, \(8u - 75=2u - 15\).

Step2: Solve the equation for \(u\)

Subtract \(2u\) from both sides: \(8u-2u - 75=2u-2u - 15\), which simplifies to \(6u-75=- 15\).
Add \(75\) to both sides: \(6u-75 + 75=-15 + 75\), so \(6u = 60\).
Divide both sides by \(6\): \(u=\frac{60}{6}=10\).

Step3: Find the value of \(FG\)

Substitute \(u = 10\) into the formula for \(FG\): \(FG=8u-75\).
\(FG=8\times10-75\).
\(FG=80 - 75\).

Answer:

\(5\)