Question

In the diagram below of triangle JKL, M is a midpoint of JK and N is a midpoint of KL . If MN=10x-11, and JL=30-6x, what is the measure of MN

Answer 1

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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,

`(MK)/(JK)=(KN)/(KL)=(MN)/(JL)`

We can use the last two ratios,

`(KN)/(KL)=(MN)/(JL)`

But, we are given the information that,

M is a midpoint of JK and N is a midpoint of KL.

Therefore,

`\begin{gathered} KN=(KL)/(2) \\ or \\ (KN)/(KL)=(1)/(2) \end{gathered}`

Thus;

`(MN)/(JL)=(1)/(2)`

MN = 10x-11 and JL= 30-6x, lets put this into the relation;

`\begin{gathered} (10x-11)/(30-6x)=(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}`

Now, we can find the measure of MN, this is;

`\begin{gathered} MN=10(2)-11 \\ =20-11 \\ =9 \end{gathered}`

Therefore, MN = 9