Question

A triangle has vertices L(-5, 6), M(-2, -3),  and N(4, 5). Write an equation in Point-Slope Form for the line perpendicular to segment LM that contains point N.

Answer 1

`\begin{gathered} \text{If we have a line} \\ y=m_1x+b, \\ \text{ Then the slope of a perpendicular line to the first must satisfy} \\ m_1m_2=-1 \\ \text{ That means that} \\ m_2=(-1)/(m_1) \end{gathered}`
`\begin{gathered} \text{ In our case, the line ML has slope} \\ m_1=(-3-6)/(-2-(-5))=(-9)/(3)=-3 \\ \text{ So, the slope of the perpendicular line to ML has slope} \\ m_2=(-1)/(m_1)=(-1)/(-3)=(1)/(3) \end{gathered}`
`\begin{gathered} \text{ Then the line perpendicular to segment ML has the equation} \\ y=(1)/(3)x+b \\ \text{ And we have that the point (4,5) must be in the line, so} \\ 5=(1)/(3)(4)\text{ + b} \\ \text{ So } \\ b=5-(4)/(3) \\ b=(15-4)/(3)=(11)/(3) \\ \\ \text{and the equation of the line perpendicular do ML and that contains N is} \\ \\ y=(1)/(3)x+(11)/(3) \end{gathered}`