Question

1. find fg

Explanation:

Step1: Use the Mid - segment Theorem

The mid - segment theorem states that the length of the mid - segment of a triangle (a segment connecting the mid - points of two sides of a triangle) is half the length of the third side. Here, \(FG\) is the mid - segment of \(\triangle JKL\) (since \(F\) is the mid - point of \(JK\) and \(G\) is the mid - point of \(KL\)). So, \(FG=\frac{1}{2}JL\).
We know that \(JK=-2x + 8\), \(JL=x + 27\), and \(FG=x + 17\).
Since \(FG=\frac{1}{2}JL\), we can also use the property of the mid - segment in terms of the side \(JK\). Another way is to use the fact that \(JK = 2FG\) (because \(F\) is the mid - point of \(JK\) and \(FG\parallel JL\)).
Substitute the expressions: \(-2x+8 = 2(x + 17)\).

Step2: Solve the equation for \(x\)

Expand the right - hand side of the equation \(-2x+8 = 2(x + 17)\):
\(-2x+8=2x + 34\).
Add \(2x\) to both sides: \(8=4x + 34\).
Subtract 34 from both sides: \(4x=8 - 34=-26\) (This is wrong. Let's start again from \(FG=\frac{1}{2}JL\)).
Since \(FG=\frac{1}{2}JL\), we have \(x + 17=\frac{1}{2}(x + 27)\).
Multiply both sides by 2 to get rid of the fraction: \(2(x + 17)=x + 27\).
Expand: \(2x+34=x + 27\).
Subtract \(x\) from both sides: \(2x - x+34=x - x + 27\), so \(x+34 = 27\).
Subtract 34 from both sides: \(x=27 - 34=-7\).

Step3: Find the length of \(FG\)

Substitute \(x=-7\) into the expression for \(FG\).
Since \(FG=x + 17\), then \(FG=-7+17\).

Answer:

\(FG = 10\)