Question 150985




We can see that the equation {{{y=(1/2)x+2}}} has a slope {{{m=1/2}}} and a y-intercept {{{b=2}}}.



Now to find the slope of the perpendicular line, simply flip the slope {{{m=1/2}}} to get {{{m=2/1}}}. Now change the sign to get {{{m=-2}}}. So the perpendicular slope is {{{m=-2}}}.



Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope {{{m=1/2}}} and the coordinates of the given point *[Tex \LARGE \left\(-4,5\right\)].



{{{y-y[1]=m(x-x[1])}}} Start with the point slope formula



{{{y-5=-2(x--4)}}} Plug in {{{m=-2}}}, {{{x[1]=-4}}}, and {{{y[1]=5}}}



{{{y-5=-2(x+4)}}} Rewrite {{{x--4}}} as {{{x+4}}}



{{{y-5=-2x+-2(4)}}} Distribute



{{{y-5=-2x-8}}} Multiply



{{{y=-2x-8+5}}} Add 5 to both sides. 



{{{y=-2x-3}}} Combine like terms. 



So the equation of the line perpendicular to {{{y=(1/2)x + 2}}} that goes through the point *[Tex \LARGE \left\(-4,5\right\)] is {{{y=-2x-3}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,(1/2)x+2,-2x-3)
circle(-4,5,0.08),
circle(-4,5,0.10),
circle(-4,5,0.12))}}}Graph of the original equation {{{y=(1/2)x+2}}} (red) and the perpendicular line {{{y=-2x-3}}} (green) through the point *[Tex \LARGE \left\(-4,5\right\)].