Question

find the lengths of \\( \overline { l b } \\), \\( \overline { b g } \\), and \\( \overline { l g } \\).
\\( l b = \\) type your answer...
\\( b g = \\) type your answer...
\\( l g = \\) type your answer...

Explanation:

Step1: Use the Mid - segment Theorem

The Mid - segment Theorem states that the mid - segment of a triangle (a segment connecting the midpoints of two sides of a triangle) is parallel to the third side and half its length.
Here, \(MB\) is the mid - segment of \(\triangle JLG\) (since \(M\) is the midpoint of \(JL\) and \(B\) is the midpoint of \(LG\)). So, \(MB=\frac{1}{2}JG\) and \(\frac{LB}{BG}=\frac{LM}{MJ}\) (by the Basic Proportionality Theorem or Thales' theorem). Also, since \(M\) is the midpoint of \(JL\) (\(LM = MJ = 6\)), then \(LB=BG\).
Set up the equation for the sides: \(6x - 2=2(2x + 1)\)

Step2: Solve the equation for \(x\)

Expand the right - hand side: \(6x-2 = 4x+2\)
Subtract \(4x\) from both sides: \(6x-4x-2=4x - 4x+2\), which gives \(2x-2 = 2\)
Add 2 to both sides: \(2x-2 + 2=2 + 2\), so \(2x=4\)
Divide both sides by 2: \(x = 2\)

Step3: Find the lengths of \(LB\), \(BG\), and \(LG\)

Substitute \(x = 2\) into the expressions:
For \(LB\): \(LB=2x + 1\), so \(LB=2\times2+1=5\)
Since \(LB = BG\), then \(BG = 5\)
For \(LG\): \(LG=LB + BG\), so \(LG=5 + 5=10\)

Answer:

\(LB = 5\), \(BG = 5\), \(LG = 10\)