# ACT Math : Perpendicular Lines

## Example Questions

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### Example Question #1 : How To Find The Equation Of A Perpendicular Line

What line is perpendicular to x + 3y = 6 and travels through point (1,5)?

y = –1/3x – 4

y = 3x + 2

y = 6x – 3

y = 2x + 1

y = 2/3x + 6

y = 3x + 2

Explanation:

Convert the equation to slope intercept form to get y = –1/3x + 2.  The old slope is –1/3 and the new slope is 3.  Perpendicular slopes must be opposite reciprocals of each other:  m1 * m2 = –1

With the new slope, use the slope intercept form and the point to calculate the intercept: y = mx + b or 5 = 3(1) + b, so b = 2

So y = 3x + 2

### Example Question #1 : How To Find The Equation Of A Perpendicular Line

What line is perpendicular to and passes through ?

Explanation:

Convert the given equation to slope-intercept form.

The slope of this line is . The slope of the line perpendicular to this one will have a slope equal to the negative reciprocal.

The perpendicular slope is .

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

So the equation of the perpendicular line is .

### Example Question #3 : How To Find The Equation Of A Perpendicular Line

What is the equation of a line that runs perpendicular to the line 2x + = 5 and passes through the point (2,7)?

2x – y = 6

x/2 + y = 6

x/2 + y = 5

x/2 – y = 6

2x + y = 7

x/2 + y = 6

Explanation:

First, put the equation of the line given into slope-intercept form by solving for y. You get y = -2x +5, so the slope is –2. Perpendicular lines have opposite-reciprocal slopes, so the slope of the line we want to find is 1/2. Plugging in the point given into the equation y = 1/2x + b and solving for b, we get b = 6. Thus, the equation of the line is y = ½x + 6. Rearranged, it is –x/2 + y = 6.

### Example Question #4 : How To Find The Equation Of A Perpendicular Line

Line m passes through the points (1, 4) and (5, 2). If line p is perpendicular to m, then which of the following could represent the equation for p?

x  y = 3

4x  3y = 4

2x  y = 3

3x + 2y = 4

2x + y = 3

2x  y = 3

Explanation:

The slope of m is equal to   y2-y1/x2-x1  =  2-4/5-1 -1/2

Since line p is perpendicular to line m, this means that the products of the slopes of p and m must be 1:

(slope of p) * (-1/2) = -1

Slope of p = 2

So we must choose the equation that has a slope of 2. If we rewrite the equations in point-slope form (y = mx + b), we see that the equation 2x  y = 3 could be written as y = 2x – 3. This means that the slope of the line 2x – y =3 would be 2, so it could be the equation of line p. The answer is 2x – y = 3.

### Example Question #1 : How To Find The Equation Of A Perpendicular Line

What is the equation for the line that is perpendicular to  through point ?

Explanation:

Perpendicular slopes are opposite reciprocals.

The given slope is found by converting the equation to the slope-intercept form.

The slope of the given line is and the perpendicular slope is  .

We can use the given point and the new slope to find the perpendicular equation. Plug in the slope and the given coordinates to solve for the y-intercept.

Using this y-intercept in slope-intercept form, we get out final equation: .

### Example Question #1 : How To Find The Equation Of A Perpendicular Line

Which line below is perpendicular to ?

Explanation:

The definition of a perpendicular line is one that has a negative, reciprocal slope to another.

For this particular problem, we must first manipulate our initial equation into a more easily recognizable and useful form: slope-intercept form or .

According to our  formula, our slope for the original line is . We are looking for an answer that has a perpendicular slope, or an opposite reciprocal. The opposite reciprocal of  is . Flip the original and multiply it by

Our answer will have a slope of . Search the answer choices for  in the  position of the equation.

(As an aside, the negative reciprocal of 4 is . Place the whole number over one and then flip/negate. This does not apply to the above problem, but should be understood to tackle certain permutations of this problem type where the original slope is an integer.)

### Example Question #1 : How To Find The Equation Of A Perpendicular Line

If a line has an equation of , what is the slope of a line that is perpendicular to the line?

Explanation:

Putting the first equation in slope-intercept form yields .

A perpendicular line has a slope that is the negative inverse. In this case, .

### Example Question #8 : How To Find The Equation Of A Perpendicular Line

Which of the following is possibly a line perpendicular to ?

Explanation:

To start, begin by dividing everything by , this will get your equation into the format .  This gives you:

Now, recall that the slope of a perpendicular line is the opposite and reciprocal slope to its mutually perpendicular line.  Thus, if our slope is , then the perpendicular line's slope must be .  Thus, we need to look at our answers to determine which equation has a slope of .  Among the options given, the only one that matches this is .  If you solve this for , you will get:

### Example Question #9 : How To Find The Equation Of A Perpendicular Line

Which of the following is the equation of a line perpendicular to the line given by:

?

Explanation:

For two lines to be perpendicular their slopes must have a product of .
and so we see the correct answer is given by

### Example Question #10 : How To Find The Equation Of A Perpendicular Line

What is the equation of a line perpendicular to the line defined by the equaiton:

Explanation:

Perpendicular lines have slopes whose product is .

Looking at our equations we can see that it is in slope-intercept form where the m value represents the slope of the line,

.

In our case we see that

therefore, .

Since

we see the only possible answer is

.

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