Given the figure, we can deduce the following information:
Point A =(-1,3)
Point B=(-2,0)
Point P=(9,-8)
To determine the line that is perpendiculat to the line segment AB and passes through point P, we first find the equation of the line segment AB by using the formula:
[tex]m=\frac{rise}{run}[/tex]
where:
rise=3
run= 1
m=slope
We plug in what we know:
[tex]m=\frac{rise}{run}=\frac{3}{1}=3[/tex]
We also note that the slopes of perpedicular lines are negative reciprocals of each other. Hence,
[tex]m_2=-\frac{1}{m_1}[/tex]
where:
m1=3
So,
[tex]\begin{gathered} m_{2}=-\frac{1}{m_{1}} \\ m_2=-\frac{1}{3} \end{gathered}[/tex]
Next, we plug in m=-1/3, x=9, and y=-8 into y=mx+b:
[tex]\begin{gathered} y=mx+b \\ -8=(-\frac{1}{3})(9)+b \\ Simplify\text{ and rearrange} \\ -8=-3+b \\ b=-8+3 \\ b=-5 \end{gathered}[/tex]
Then, we plug in m=-1/3 and b=-5, into y=mx+b:
[tex]\begin{gathered} y=mx+b \\ y=-\frac{1}{3}x-5 \end{gathered}[/tex]
We can conclude that the equation of the line is y=-1/3x-5. Therefore, the line that is perpendicular to the line segment AB that passes through point P is shown in the graph below:
