Question

In the diagram below of triangle NPQ, R is a midpoint of NP and S is a midpoint of PQ. If RS 15 - x, and NQ = 9x - 36, what is the measure of NQ?

Answer 1

The triangle midpoint theorem is as stated above.

In our case,

RS is joining the midpoints of NP and PQ.

Hence by the triangle midpoint theorem,

$\begin{gathered} RS\parallel NQ\text{ and } \\ RS=\frac{1}{2}NQ \end{gathered}$

Therefore,

triangle PRS is similar to triangle PNQ.

This means that the ratios of their corresponding sides are equal.

$\frac{NQ}{RS}=\frac{NP}{RP}$

Since R is the midpoint of NP then

$\frac{NP}{RP}=2$

Therefore,

$\begin{gathered} \frac{NQ}{RS}=2 \\ \Rightarrow NQ=2RS \end{gathered}$

Hence,

$\begin{gathered} 9x-36=2(15-x) \\ \Rightarrow9x-36=30-2x \\ \Rightarrow9x+2x=30+36 \\ \Rightarrow11x=66 \\ \Rightarrow x=\frac{66}{11}=6 \end{gathered}$$\begin{gathered} \text{ Therefore,} \\ NQ=9x-36 \\ \text{gives} \\ NQ=9(6)-36=54-36=18 \end{gathered}$

Hence the measure of NQ is 18