Question

question
in the diagram below of triangle ( j k l, m ) is the midpoint of ( overline{j l} ) and ( n ) is the midpoint of ( overline{k l} ). if ( m n=-3+2 x ), and ( j k=29-3 x ), what is the measure of ( overline{m n} )?

Explanation:

Step1: Apply the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment (\(MN\)) of a triangle is half the length of the third side (\(JK\)). So, \(MN=\frac{1}{2}JK\).

Step2: Substitute the given expressions

Given \(MN = - 3+2x\) and \(JK = 29 - 3x\), we substitute into the equation \(MN=\frac{1}{2}JK\). So, \(-3 + 2x=\frac{1}{2}(29 - 3x)\).

Step3: Solve the equation for \(x\)

Multiply both sides of the equation \(-3 + 2x=\frac{1}{2}(29 - 3x)\) by \(2\) to get \(2(-3 + 2x)=29 - 3x\).
Expand the left - hand side: \(-6 + 4x=29 - 3x\).
Add \(3x\) to both sides: \(-6+4x + 3x=29-3x + 3x\), which simplifies to \(-6 + 7x=29\).
Add \(6\) to both sides: \(-6+6 + 7x=29 + 6\), so \(7x=35\).
Divide both sides by \(7\): \(x = 5\).

Step4: Find the length of \(MN\)

Substitute \(x = 5\) into the expression for \(MN\): \(MN=-3+2x\).
\(MN=-3+2\times5=-3 + 10=11\).

Answer:

\(11\)