Geometry - Perpendicular Lines

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

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

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:

Which line is perpendicular to the given line below?

$y=-\frac{5}{2}x-4$

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

$y=-\frac{5}{2}x+4$

$y=-\frac{2}{5}x$

$y=\frac{5}{2}x-4$

Explanation

Two perpendicular lines have slopes that are opposite reciprocals, meaning that the sign changes and the reciprocal of the slope is taken.

The original equation is in slope-intercept form,

where represents the slope.

In this case, the slope of the original is:

$m=-\frac{5}{2}$

After taking the opposite reciprocal, the result is the slope below:

$m=\frac{2}{5}$

The slopes of two lines on the coordinate plane are and 4.

True or false: the lines are perpendicular.

True

False

Explanation

Two lines on the coordinate plane are perpendicular if and only if the product of their slopes is . The product of the slopes of the lines in question is

$- 0.25 \cdot 4 = -1$ ,

so the lines are indeed perpendicular.

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:

The slopes of two lines on the coordinate plane are and 4.

True or false: the lines are perpendicular.

True

False

Explanation

Two lines on the coordinate plane are perpendicular if and only if the product of their slopes is . The product of the slopes of the lines in question is

$- 0.25 \cdot 4 = -1$ ,

so the lines are indeed perpendicular.

Two lines intersect at the point . One line passes through the point ; the other passes through .

True or false: The lines are perpendicular.

False

True

Explanation

Two lines are perpendicular if and only if the product of their slopes is . The slope of each line can be found from the coordinates of two points using the slope formula

$m = \frac{y_{2}- y_{1}}{x_{2}-x_{1}}$

To find the slope of the first line, set $x_{1} = 3,y_{1}= 8 , x_{2} = 7, y_{2}= 3$ :

$m = \frac{3-8}{7-3} = \frac{-5}{4} = -\frac{5}{4}$

To find the slope of the second line, set $x_{1} = 3,y_{1}= 8 , x_{2} = -2, y_{2}=1 2$ :

$m = \frac{12-8}{-2-3} = \frac{4}{-5} = -\frac{4}{5}$

The product of the slopes is

$-\frac{5}{4} \left ( -\frac{4}{5} \right )= 1$

As the product is not , the lines are not perpendicular.

Which line is perpendicular to the given line below?

$y=-\frac{5}{2}x-4$

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

$y=-\frac{5}{2}x+4$

$y=-\frac{2}{5}x$

$y=\frac{5}{2}x-4$

Explanation

Two perpendicular lines have slopes that are opposite reciprocals, meaning that the sign changes and the reciprocal of the slope is taken.

The original equation is in slope-intercept form,

where represents the slope.

In this case, the slope of the original is:

$m=-\frac{5}{2}$

After taking the opposite reciprocal, the result is the slope below:

$m=\frac{2}{5}$

Two lines intersect at the point . One line passes through the point ; the other passes through .

True or false: The lines are perpendicular.

False

True

Explanation

Two lines are perpendicular if and only if the product of their slopes is . The slope of each line can be found from the coordinates of two points using the slope formula

$m = \frac{y_{2}- y_{1}}{x_{2}-x_{1}}$

To find the slope of the first line, set $x_{1} = 3,y_{1}= 8 , x_{2} = 7, y_{2}= 3$ :

$m = \frac{3-8}{7-3} = \frac{-5}{4} = -\frac{5}{4}$

To find the slope of the second line, set $x_{1} = 3,y_{1}= 8 , x_{2} = -2, y_{2}=1 2$ :

$m = \frac{12-8}{-2-3} = \frac{4}{-5} = -\frac{4}{5}$

The product of the slopes is

$-\frac{5}{4} \left ( -\frac{4}{5} \right )= 1$

As the product is not , the lines are not perpendicular.

Perpendicular Lines

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