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ACT Math : How to find the equation of a perpendicular line

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Example Question #11 : How To Find The Equation Of A Perpendicular Line

Line follows the equation $y=\frac{3}{4}x -2$ .

State the equation, in slope-intercept form, of line , which is perpendicular to line and intersects it at point (24, 18) .

Possible Answers:

$y=-\frac{4}{3}x + 14$

$y=\frac{4}{3}x + 42$

$y=\frac{3}{4}x -6$

$y=-\frac{4}{3}x + 50$

Correct answer:

$y=-\frac{4}{3}x + 50$

Explanation:

Since line follows the equation $y=\frac{3}{4}x -2$ , we can surmise its slope is $m=\frac{3}{4}$ . Thus, it follows that any line perpendicular to will have a slope of $n = -\frac{1}{m} = -\frac{1}{(\frac{3}{4})} = -\frac{4}{3}$ .

Since we also know at least one point on line , (24, 18) , we can use point slope form to find an initial equation for our line.

y-y_1=m(x-x_1) ----> $y-18=-\frac{4}{3}(x-24)$ .

Next, we can simplify to reach point-slope form.

$y-18=-\frac{4}{3}(x-24)$ ----> $y=-\frac{4}{3}x + 50$ .

Thus, line in slope-intercept form is $y=-\frac{4}{3}x + 50$ .

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Example Question #11 : How To Find The Equation Of A Perpendicular Line

Which of the following equations represents a line that goes through the point (-2, 6) and is perpendicular to the line 3x + 6y = 12 ?

Possible Answers:

y = -2x + 10

y = 2x + 10

y = 2x + 2

y = 2x + 6

y = -2x + 2

Correct answer:

y = 2x + 10

Explanation:

In order to solve this problem, we need first to transform the equation from standard form to slope-intercept form:

y = mx + b

Transform the original equation to find its slope.

3x + 6y = 12

First, subtract from both sides of the equation.

3x-3x + 6y = 12-3x

Simplify and rearrange.

6y = -3x + 12

Next, divide both sides of the equation by 6.

$\frac{6y}{6} = -\frac{3x}{6} + \frac{12}{6}$

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

The slope of our first line is equal to $-\frac{1}{2}$ . Perpendicular lines have slopes that are opposite reciprocals of each other; therefore, if the slope of one is x, then the slope of the other is equal to the following:

$-\frac{1}{x}$

Let's calculate the opposite reciprocal of our slope:

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

The slope of our line is equal to 2. We now have the following partial equation:

y = 2x + b

We are missing the y-intercept, . Substitute the x- and y-values in the given point (-2, 6) to solve for the missing y-intercept.

6 = 2(-2) + b

6 = -4 + b

Add 4 to both sides of the equation.

6+4 = -4+4 + b

10 = b

Substitute this value into our partial equation to construct the equation of our line:

y = 2x + 10

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ACT Math : How to find the equation of a perpendicular line

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Example Question #11 : How To Find The Equation Of A Perpendicular Line

Example Question #11 : How To Find The Equation Of A Perpendicular Line

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