Question

Which point is on the line that passes through point Z and is perpendicular to line AB?
A. (-4,1)
B. (1,-2)
C. (2,0)
D. (4,4)

Answer 1

bearing in mind that perpendicular lines have negative reciprocal slopes, hmmm what is the slope of AB anyway?

$\bf A(\stackrel{x_1}{-2}~,~\stackrel{y_1}{4})\qquad B(\stackrel{x_2}{0}~,~\stackrel{y_2}{-4}) \\\\\\ % slope = m slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{-4-4}{0-(-2)}\implies \cfrac{-4-4}{0+2}\implies -4$

so the perpendicular line to point Z will have a slope of

$\bf \stackrel{\textit{perpendicular lines have \underline{negative reciprocal} slopes}} {-4\implies \stackrel{slope}{\cfrac{-4}{1}}\qquad \qquad \qquad \stackrel{reciprocal}{-\cfrac{1}{4}}\qquad \stackrel{negative~reciprocal}{\cfrac{1}{4}}}$

so, the equation of such line will then be

$\bf Z(\stackrel{x_1}{0}~,~\stackrel{y_1}{2})\qquad \qquad slope = m\implies \cfrac{1}{4} \\\\\\ % point-slope intercept \stackrel{\textit{point-slope form}}{y- y_1= m(x- x_1)}\implies y-2=\cfrac{1}{4}(x-0)\implies y-2=\cfrac{1}{4}x$

and now, let's test those points,

$\bf (\stackrel{x}{-4}~~,~~\stackrel{y}{1})\qquad \qquad \boxed{1}-2=\cfrac{1}{4}\boxed{-4}\implies -1=\cfrac{-4}{4}\implies -1=-1\quad \checkmark$

Answer 2

The point is on the line that passes through point Z and is perpendicular to line AB is (-4,1) and this can be determined by using the slope-intercept form of the line.

Given :

Points  ---  A(-2,4), B(0,-4), and Z(0,2)

First, determine the slope of line AB. The slope of line AB is given by:

$\rm m = \dfrac{-4-4}{0+2}$

m = -4

The slope of the line which is perpendicular to the line AB is given by:

mm' = -1

m' = 1/4

Now, the equation of the line perpendicular to the line AB is given by:

$(y - 2) = \dfrac{1}{4}(x-0)$

4y = x + 8   --- (1)

Now, the point which satisfies the equation (1) is (-4,1). Therefore, the correct option is A).

For more information, refer to the link given below:

https://brainly.com/question/11824567