How to find the equation of a perpendicular line

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Math › How to find the equation of a perpendicular line

Questions 1 - 10

What line is perpendicular to $y=\frac{1}{2}x+3$ through ?

Explanation

Perdendicular lines have slopes that are opposite reciprocals. The slope of the old line is $\frac{1}{2}$ , so the new slope is .

Plug the new slope and the given point into the slope intercept equation to calculate the intercept:

or , so .

Thus , or .

Suppose a line is represented by a function . Find the equation of a perpendicular line that intersects the point .

$f(x)= -\frac{1}{2}x+2\frac{1}{2}$

$f(x)= -\frac{1}{2}x+1\frac{1}{2}$

$f(x)= \frac{1}{2}x-1\frac{1}{2}$

$f(x)= -\frac{1}{2}x$

Explanation

Determine the slope of the function . The slope is:

The slope of a perpendicular line is the negative reciprocal of the original slope. Determine the value of the slope perpendicular to the original function.

$m2 = - \frac{1}{m1} = -\frac{1}{2}$

Plug in the given point and the slope to the slope-intercept form to find the y-intercept.

$2=\left(-\frac{1}{2}\right)(1)+b$

$b=2\frac{1}{2}$

Substitute the slope of the perpendicular line and the new y-intercept back in the slope-intercept equation, .

The correct answer is: $f(x)= -\frac{1}{2}x+2\frac{1}{2}$

Write an equation in slope-intercept form for the line that passes through $\left ( 3,1 \right )$ and that is perpendicular to a line which passes through the two points $\left ( 3,1 \right )$ and $\left ( -2,5 \right )$ .

$y=\frac{5}{4}X-\frac{11}{4}$

$y=-\frac{4}{5}X+11$

$y=\frac{5}{4}X+\frac{11}{4}$

Explanation

Find the slope of the line through the two points. It is $-\frac{4}{5}$ .

Since the slope of a perpendicular line is the negative reciprocal of the original line, the new line's slope is $\frac{5}{4}$ . Plug the slope and one of the points into the point-slope formula $y-y_{1}=m(x-x_{1})$ . Isolate for .

Which of the following lines is perpendicular to the following line?

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

$y=-\frac{1}{2}x$

Explanation

Perpendicular lines have slopes that are opposite reciprocals meaning that the slope is flpped. The equation that satisfies both of these criteria is:

Find the equation of a line perpendicular to

$y=\frac{1}{3}X+8$

Explanation

Since a perpendicular line has a slope that is the negative reciprocal of the original line, the new slope is $\frac{1}{3}$ . There is only one answer with the correct slope.

What is a line that is perpendicular to ?

$y=-\frac{1}3x+4{}$

$2-y=\frac{1}{12}x-1$

Explanation

A line is perpendicular to another line when they meet at a degree angle. That angle is the result of the slopes of the lines being opposite reciprocals.

The "opposite reciprocal" of is best described as $-\frac{1}{m}$ .

First we reorganize the original equation to isolate . To do this we want to get our equation into slope intercept form .

First subtract 12 from each side.

Now divide by 4 and simplify where possible.

$y=3x-\frac{13}{4}$

opposite reciprocal $=-\frac{1}{m}=-\frac{1}{3}$

The only equation in the answer choices with a slope of $-\frac{1}{3}$ is

$y=-\frac{1}3x+4{}$ .

A line is perpendicular to the line of the equation

and passes through the point .

Give the equation of the line.

$\textup{None of the other choices gives the correct response}$

Explanation

A line perpendicular to another line will have as its slope the opposite of the reciprocal of the slope of the latter. Therefore, it is necessary to find the slope of the line of the equation

Rewrite the equation in slope-intercept form . , the coefficient of , will be the slope of the line.

Add to both sides:

Multiply both sides by $\frac{1}{7}$ , distributing on the right:

$\frac{1}{7} \cdot 7y = \frac{1}{7} \cdot( -5x+32)$

$y = -\frac{5}{7}x+\frac{32}{7}$

The slope of this line is $-\frac{5}{7}$ . The slope of the first line will be the opposite of the reciprocal of this, or $\frac{7}{5}$ . The slope-intercept form of the equation of this line will be

$y = \frac{7}{5}x+b$ .

To find , set and and solve:

$8 = \frac{7}{5} (-2)+b$

$8 = - \frac{14}{5} +b$

$\frac{40}{5} = - \frac{14}{5} +b$

Add $\frac{14}{5}$ to both sides:

$\frac{40}{5} + \frac{14}{5} = - \frac{14}{5} +b + \frac{14}{5}$

$\frac{54}{5} = b$

The equation, in slope-intercept form, is $y = \frac{7}{5}x+ \frac{54}{5}$ .

To rewrite in standard form with integer coefficients:

Multiply both sides by 5:

$5 y =5 \left (\frac{7}{5}x+ \frac{54}{5} \right )$

$5 y =5 \left (\frac{7}{5}x \right ) + 5 \left ( \frac{54}{5} \right )$

Add to both sides:

What is the slope of a line perpendicular to the line with the following equation?

$3y+5x=9$

$\frac{3}{5}$

$-\frac{3}{5}$

$\frac{5}{3}$

$-\frac{5}{3}$

Answer cannot be determined from this information.

Explanation

Step 1: get the line into y = mx +b format to find the slope m:

$\3x+5y=9\rightarrow \\ 3y=-5x+9\rightarrow\\ y=-\frac{5}{3}x+3$

Next, we need to remember that perpendicular lines have slopes that are negative reciprocals. Find the negative reciprocal of $-\frac{5}{3}$ :

$\frac{3}{5}$

Suppose a perpendicular line passes through and point . Find the equation of the perpendicular line.

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

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

$y=\frac{1}{2}x+5\frac{1}{2}$

Explanation

Find the slope from the given equation . The slope is: .

The slope of the perpendicular line is the negative reciprocal of the original slope.

$m2=-\frac{1}{m1}=-\frac{1}{2}$

Plug in the perpendicular slope and the given point to the slope-intercept equation.

$5=\left(-\frac{1}{2}\right)(-1)+b$

$5=\frac{1}{2}+b$

$b=4\frac{1}{2}$

Plug in the perpendicular slope and the y-intercept into the slope-intercept equation to get the equation of the perpendicular line.

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

What is the equation of a line that is perpendicular to and passes through point ?

$y=-\frac{1}{4}x+3$

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

Explanation

When finding the slope of a perpendicular line, we need to ensure we have form. stands for slope. Our is . To find the perpendicular slope, we need to take the negative reciprocal of that value which is $-\frac{1}{4}$ . Since we are looking for an equation, we need to reuse the form to solve for . We do this by plugging in our coordinates.