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Sagot :
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The triangles MKN and JKL are therefore similar triangles, we can use similarity relations to obtain their measures.
These similarity measures apply,
[tex]\frac{MK}{JK}=\frac{KN}{KL}=\frac{MN}{JL}[/tex]We can use the last two ratios,
[tex]\frac{KN}{KL}=\frac{MN}{JL}[/tex]But, we are given the information that,
M is a midpoint of JK and N is a midpoint of KL.
Therefore,
[tex]\begin{gathered} KN=\frac{KL}{2} \\ or \\ \frac{KN}{KL}=\frac{1}{2} \end{gathered}[/tex]Thus;
[tex]\frac{MN}{JL}=\frac{1}{2}[/tex]MN = 10x-11 and JL= 30-6x, lets put this into the relation;
[tex]\begin{gathered} \frac{10x-11}{30-6x}=\frac{1}{2} \\ \text{cross multiply} \\ 2(10x-11)=30-6x \\ 20x-22=30-6x \\ \text{collect like terms} \\ 20x+6x=30+22 \\ 26x=52 \\ \text{divide both sides by 2}6 \\ x=2 \end{gathered}[/tex]Now, we can find the measure of MN, this is;
[tex]\begin{gathered} MN=10(2)-11 \\ =20-11 \\ =9 \end{gathered}[/tex]Therefore, MN = 9
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