Question

4 (overrightarrow{df}) bisects (angle edg). find (fg).

Explanation:

Step1: Use the Angle Bisector Theorem

Since \( \overrightarrow{DF} \) bisects \( \angle EDG \), and \( FE\perp DE\), \( FG\perp DG \), by the Angle Bisector Theorem (a point on the bisector of an angle is equidistant from the sides of the angle), we have \( FE = FG \). So, \( n + 8=3n - 4 \).

Step2: Solve the equation for \(n\)

Subtract \(n\) from both sides: \(n + 8 - n=3n - 4 - n\), which gives \(8 = 2n-4\).
Add \(4\) to both sides: \(8 + 4=2n-4 + 4\), so \(12 = 2n\).
Divide both sides by \(2\): \(n=\frac{12}{2}=6\).

Step3: Find the length of \(FG\)

Substitute \(n = 6\) into the expression for \(FG\) (\(FG = 3n - 4\)). Then \(FG=3\times6 - 4\).
Calculate \(3\times6 - 4=18 - 4 = 14\).

Answer:

\(14\)