Question

in the diagram below of triangle ( jkl ), ( m ) is the midpoint of ( overline{jl} ) and ( n ) is the midpoint of ( overline{kl} ). if ( mn = 16 + 2x ), and ( jk = 3x + 34 ), what is the measure of ( overline{jk} )?

Explanation:

Step1: Use the mid - segment theorem

In \(\triangle JKL\), since \(M\) is the mid - point of \(\overline{JL}\) and \(N\) is the mid - point of \(\overline{KL}\), by the mid - segment theorem, \(MN=\frac{1}{2}JK\).

Step2: Substitute the given expressions

Given \(MN = 16 + 2x\) and \(JK=3x + 34\). Substituting into \(MN=\frac{1}{2}JK\), we get \(16 + 2x=\frac{1}{2}(3x + 34)\).

Step3: Solve the equation

Multiply both sides of the equation \(16 + 2x=\frac{1}{2}(3x + 34)\) by \(2\) to get \(32+4x = 3x + 34\).
Subtract \(3x\) from both sides: \(32+4x-3x=3x + 34-3x\), which simplifies to \(32+x=34\).
Subtract \(32\) from both sides: \(x=34 - 32=2\).

Step4: Find the length of \(JK\)

Substitute \(x = 2\) into \(JK = 3x+34\). Then \(JK=3\times2+34=6 + 34=70\).

Answer:

\( 70 \)