Question

find the missing length indicated.

1. find lm

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, \(LM\) is the mid - segment of \(\triangle TUV\) since \(L\) is the mid - point of \(TU\) and \(M\) is the mid - point of \(UV\). So, \(LM=\frac{1}{2}TV\).

Step2: Set up the equation

We know that \(LM = x + 4\) and \(TV=3x + 2\). Substituting into the mid - segment formula: \(x + 4=\frac{1}{2}(3x + 2)\).
Multiply both sides by 2 to get rid of the fraction: \(2(x + 4)=3x+2\).
Expand the left - hand side: \(2x+8 = 3x + 2\).
Subtract \(2x\) from both sides: \(8=x + 2\).
Subtract 2 from both sides: \(x=6\).

Step3: Find the length of \(LM\)

Substitute \(x = 6\) into the expression for \(LM\). Since \(LM=x + 4\), then \(LM=6 + 4\).

Answer:

\(10\)