Side Spitter Theorem : If a line intersects two sides of a triangle and is parallel to the third side of the triangle, it divides those two sides proportionally.
In the given triangle EOG, the line MT cuts the triangle:
According to the side split theorem :
[tex]\frac{EM}{MG}=\frac{ET}{TO}[/tex]Substitute the value and solve for x :
[tex]\begin{gathered} \frac{EM}{MG}=\frac{ET}{TO} \\ \frac{12}{6}=\frac{3x-2}{x+2} \\ \text{Apply cross multiplication:} \\ \frac{2}{1}=\frac{3x-2}{x+2} \\ 2(x+2)=3x-2 \\ 2x+4=3x-2 \\ 3x-2x=4+2 \\ x=6 \end{gathered}[/tex]x = 6
Substitute x = 6 in the expression of ET = 3x - 2
ET = 3x - 2
ET = 3(6) - 2
ET = 18 - 2
ET = 16
Answer : The length of the line segment is ET = 16
