Question

in the diagram below of triangle ijk, l is the mid - point of ik and m is the mid - point of jk. if lm = 8x - 53, and ij=-26 + 6x, what is the measure of lm?

Explanation:

Step1: Recall mid - segment theorem

The mid - segment of a triangle (the line segment joining the mid - points of two sides of a triangle) is parallel to the third side and half its length. So, $LM=\frac{1}{2}IJ$.

Step2: Set up the equation

Given $LM = 8x−53$ and $IJ=-26 + 6x$, we have the equation $8x−53=\frac{1}{2}(-26 + 6x)$.

Step3: Solve the equation for x

Multiply both sides of the equation by 2 to get $2(8x−53)=-26 + 6x$.
Expand the left - hand side: $16x-106=-26 + 6x$.
Subtract $6x$ from both sides: $16x-6x-106=-26+6x - 6x$, which simplifies to $10x-106=-26$.
Add 106 to both sides: $10x-106 + 106=-26+106$, so $10x = 80$.
Divide both sides by 10: $x = 8$.

Step4: Find the measure of LM

Substitute $x = 8$ into the expression for $LM$: $LM=8x−53$.
$LM=8\times8−53$.
$LM = 64−53$.
$LM = 25$.

Answer:

$25$