Question

use the image to answer the question.

in quadrilateral fgkj, \\( \overline { f k } \\) is a perpendicular bisector of \\( \overline { g j } \\). the length of segment \\( \overline { g f } \\) is \\( 6 + x \\) units, and the length of segment \\( \overline { j f } \\) is \\( 3 x - 5 \\) units. what is the value of x?

(1 point)

the value of x is \\( \square \\).

Explanation:

Step1: Use the property of perpendicular bisector

Since \( \overline{FK} \) is a perpendicular bisector of \( \overline{GJ} \), by the property of a point on the perpendicular bisector of a segment (a point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment), we have \( GF = JF \).

Step2: Set up the equation

Given \( GF=6 + x \) and \( JF = 3x-5 \), then \( 6 + x=3x - 5 \).

Step3: Solve the equation for \( x \)

Subtract \( x \) from both sides: \( 6=3x - 5-x \), which simplifies to \( 6 = 2x-5 \).
Add 5 to both sides: \( 6 + 5=2x \), so \( 11 = 2x \).
Divide both sides by 2: \( x=\frac{11}{2}=5.5 \).

Answer:

\( 5.5 \)