Question

5 fill in the blank 1 point
consider the triangle below.
find the lengths of \\( \overline { l b } \\), \\( \overline { b g } \\), and \\( \overline { l g } \\).
\\( l b = \\)
\\( b g = \\)
\\( l g = \\)

Explanation:

Step1: Use the property of the mid - segment of a triangle

Since \(MB\) is a mid - segment of \(\triangle LGJ\) (the red arrows indicate the mid - segment property), we know that \(\frac{LB}{LG}=\frac{LM}{LJ}\). Also, by the mid - segment theorem, \(\frac{LB}{BG}=\frac{LM}{MJ}\). And since \(MB\) is a mid - segment, \(\frac{LB}{BG}=\frac{LM}{MJ} = \frac{8}{12}=\frac{2}{3}\). So, \(\frac{2x + 2}{5x-1}=\frac{2}{3}\).

Step2: Cross - multiply the equation

Cross - multiplying the equation \(\frac{2x + 2}{5x-1}=\frac{2}{3}\) gives us \(3(2x + 2)=2(5x - 1)\).
Expanding both sides: \(6x+6 = 10x-2\).

Step3: Solve for \(x\)

Subtract \(6x\) from both sides: \(6=10x - 2-6x\), which simplifies to \(6 = 4x-2\).
Add \(2\) to both sides: \(6 + 2=4x\), so \(8 = 4x\).
Divide both sides by \(4\): \(x = 2\).

Step4: Find the length of \(LB\)

Substitute \(x = 2\) into \(LB=2x + 2\). Then \(LB=2\times2+2=6\).

Step5: Find the length of \(BG\)

Substitute \(x = 2\) into \(BG = 5x-1\). Then \(BG=5\times2-1=9\).

Step6: Find the length of \(LG\)

Since \(LG=LB + BG\), \(LG=6 + 9=15\).

Answer:

\(LB = 6\)
\(BG = 9\)
\(LG = 15\)