In quadrilateral FGKJ , FK¯¯¯¯¯¯¯¯ is a perpendicular bisector of GJ¯¯¯¯¯¯¯ . The length of segment GF¯¯¯¯¯¯¯¯ is 6+x units, and the length of segment JF¯¯¯¯¯¯¯ is 3x−5 units. What is the value of x?

Answer 1

Since \( FK \) is the perpendicular bisector of \( GJ \), the distances from point \( F \) to points \( G \) and \( J \) must be equal. Therefore, we can set up the equation based on the lengths given:

\[ GF = JF \]

Substituting the provided lengths into the equation, we get:

\[ 6 + x = 3x - 5 \]

Now, we will solve for \( x \):

1. Start by rearranging the equation to move all terms involving \( x \) to one side:

\[ 6 + x + 5 = 3x \]

This simplifies to:

\[ 11 + x = 3x \]

2. Now, isolate \( x \) by subtracting \( x \) from both sides:

\[ 11 = 3x - x \]

This simplifies further to:

\[ 11 = 2x \]

3. Divide both sides by 2 to solve for \( x \):

\[ x = \frac{11}{2} = 5.5 \]

Thus, the value of \( x \) is:

\[ \boxed{5.5} \]