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=(1)/(2)NQ \end{gathered}`

Therefore,

triangle PRS is similar to triangle PNQ.

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

`(NQ)/(RS)=(NP)/(RP)`

Since R is the midpoint of NP then

`(NP)/(RP)=2`

Therefore,

`\begin{gathered} (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=(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