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

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Example Questions

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Example Question #45 : Lines

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

Possible Answers:

y = -2x + 2

y = 2x + 10

y = 2x + 2

y = 2x + 6

y = -2x + 10

Correct answer:

y = 2x + 10

Explanation:

We must must transform the standard form equation 3x + 6y = 12 into a slope-intercept form equation (y = mx + b) to find its slope.

3x + 6y = 12 (Subtract 3x on both sides.)

6y = -3x + 12 (Divide both sides by 6.)

y = -3/6x + 12/6

y = -1/2x + 2

The slope of our first line is equal to -1/2. Perpendicular lines have negative reciprocal slopes, so if the slope of one is x, the slope of the other is -1/x.

The negative reciprocal of -1/2 is equal to 2, so that is the slope of our line.

We now have y = 2x + b. We plug in the point (-2, 6) and solve for b.

6 = 2(-2) + b

6 = -4 + b (Add 4 on both sides)

10 = b

The equation of our new line is y = 2x + 10

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)?

Possible Answers:

y = 6x – 3

y = 2/3x + 6

y = 2x + 1

y = –1/3x – 4

y = 3x + 2

Correct answer:

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 ?

Possible Answers:

Correct answer:

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 #2 : 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)?

Possible Answers:

x/2 – y = 6

2x – y = 6

2x + y = 7

x/2 + y = 6

x/2 + y = 5

Correct answer:

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 #1 : 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?

Possible Answers:

3x + 2y = 4

2x  y = 3

x  y = 3

4x  3y = 4

2x + y = 3

Correct answer:

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

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

Possible Answers:

Correct answer:

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 ?

Possible Answers:

Correct answer:

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.

is our answer. 

(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 2y=3x+3, what is the slope of a line that is perpendicular to the line?

Possible Answers:

-\frac{2}{3}

-2

\frac{3}{2}

-\frac{3}{2}

3

Correct answer:

-\frac{2}{3}

Explanation:

Putting the first equation in slope-intercept form yields y=\frac{3}{2}x+\frac{3}{2}.

A perpendicular line has a slope that is the negative inverse. In this case, -\frac{2}{3}.

Example Question #5 : How To Find The Equation Of A Perpendicular Line

Which of the following is possibly a line perpendicular to ?

Possible Answers:

Correct answer:

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 #3 : 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:

?

Possible Answers:

Correct answer:

Explanation:

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

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