Question

Which point is on the line that passes through point Z and is perpendicular to line AB?

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

Step 1

Find the slope of the line AB

we know that

the formula to find the slope between two points is equal to

$m=\frac{(y2-y1)}{(x2-x1)}$  

we have

$A(-2,4)\\B(0,-4)$

substitute in the formula

$m=\frac{(-4-4)}{(0+2)}$  

$m=\frac{(-8)}{(0+2)}$  

$m1=-4$  

Step 2

Find the slope of the line that passes through point Z and is perpendicular to line AB

we know that

If two lines are perpendicular, then the product of their slopes is equal to minus one

so

$m1*m2=-1$

$m2=-\frac{1}{m1}$

we have

$m1=-4$  

substitute

$m2=\frac{1}{4}$

Step 3

Find the equation of the line that passes through point Z and is perpendicular to line AB

we have

$Z(0,2)\\m2=\frac{1}{4}$

we know that

the equation of the line into point -slope form is equal to

$y-y1=m*(x-x1)$

substitute

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

$y=\frac{1}{4}x+2$

Verify which point is on the line that passes through point Z and is perpendicular to line AB

we know that  

If a point is on the line, then the coordinates of the point must be satisfy the equation of the line

we are going to proceed to verify each point to determine the solution

Substitute the values of x and y of the point in the equation of the line. If the equation is true, then the point is on the line

Step 4

point $A(-4,1)$

$1=\frac{1}{4}*(-4)+2$

$1=1$ --------> the equation is true

therefore

the point $A(-4,1)$ is on the line

Step 5

point $B(1,-2)$

$-2=\frac{1}{4}*(1)+2$

$-2=\frac{9}{4}$ --------> the equation is not true

therefore

the point $B(1,-2)$ is not on the line

Step 6

point $C(2,0)$

$0=\frac{1}{4}*(2)+2$

$0=\frac{5}{2}$ --------> the equation is not true

therefore

the point $C(2,0)$ is not on the line

Step 7

point $D(4,4)$

$4=\frac{1}{4}*(4)+2$

$4=3$ --------> the equation is not true

therefore

the point $D(4,4)$ is not on the line

the answer is the option

$A(-4,1)$