Question

1. \\( \overrightarrow { d q } \\) bisects \\( \angle e d g \\). find the value of \\( x \\).

Explanation:

Step1: Apply the Angle Bisector Theorem

According to the Angle Bisector Theorem, if a ray bisects an angle of a triangle and is perpendicular to the sides, the lengths of the perpendicular segments from a point on the bisector to the sides of the angle are equal. So, \(8x + 42=15x\).

Step2: Solve the equation

Subtract \(8x\) from both sides: \(42 = 15x-8x\).
Simplify: \(42 = 7x\).
Divide both sides by \(7\): \(x=\frac{42}{7}=6\).

Answer:

\(6\)