Question

In the figure, M and N are midpoints of RT and ST. What is the length of MN?

https://media.glynlyon.com/g_geo_2012/9/groupi94.gif

Answer 1

(a.) a + b
(b.) b - a
(c.) 2

Answer 2

First, note that the midpoint of some point \(P\) with coordinates \((x, y)\) is, by definition:
\[
P_{mid}=\left(\frac{x}{2}, \frac{y}{2}\right)
\]Hence, the midpoint of \(ST\) becomes:
\[
ST_{mid}=\left(\frac{2c+2b}{2}, \frac{2d}{2}\right)=\left(b+c, d\right)
\]A similar application to \(RT\) gives us:
\[
RT_{mid}=(a+c, d)
\]But, note that \(MN\) is parallel to the x-axis, as \(d=d\). Hence, its length becomes:
\[
MN=|ST_m-RT_m|=(b+c,d)-(a+c,d)=|(b-a, 0)|=b-a
\]