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

Study concepts, example questions & explanations for SAT Math

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Example Questions

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

Solve the system of equations for the point of intersection.

 

 

Possible Answers:

Correct answer:

Explanation:

First one needs to use one of the two equations to substitute one of the unknowns.

From the second equation we can derive that y = x – 3.

Then we substitute what we got into the first equation which gives us: x + x – 3 = 15.

Next we solve for x, so 2x = 18 and x = 9.

x – y = 3, so = 6.

These two lines will intersect at the point (9,6).

                 

 

 

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

Line A is perpendicular to y=2x+1 and passes the point (0,1). Find the -intercept of line A.

Possible Answers:

1

1.5

2.5

2

0.5

 

Correct answer:

2

Explanation:

We are given an equation of a line and told that line A is perpendicular to it.  The slope of the given line is 2.  Therefore, the slope of line A must be -0.5, since perpendicular lines have slopes that are negative reciprocals of each other.

The equation for line A will therefore take the form , where b is the y-intercept.

Since we are told that it crosses (0,1), we can plug in the point and solve for c:

Then the equation becomes y=-0.5x+1.

To find the x-intercept, plug in 0 for y and solve for x:

x=2

Example Question #13 : How To Find The Equation Of A Perpendicular Line

What line is perpendicular to  through ?

Possible Answers:

Correct answer:

Explanation:

The slope of the given line is , and the slope of the perpendicular line is its negative reciprocal, .  We take the new slope and the given point  and plug them into the slope-intercept form of a line, .

 

Thus, the perpendicular line has the equation , or in standard form, .

Example Question #14 : How To Find The Equation Of A Perpendicular Line

In the xy-plane, the equation of the line n is –x+8y=17. If the line m is perpendicular to line n, what is a possible equation of line m?

 

Possible Answers:

y= -8x + 5

y= 8x-17

y= -1/8x + 5

x= -8y + (17/8)

Correct answer:

y= -8x + 5

Explanation:

We start by add x to the other side of the equation to get the y by itself, giving us 8y =17 + x. We then divide both sides by 8, giving us y= 17/8 + 1/8x. Since we are looking for the equation of a perpendicular line, we know the slope (the coefficient in front of x) will be the opposite reciprocal of the slope of our line, giving us y= -8x + 5 as the answer.

 

 

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 = –1/3x – 4

y = 3x + 2

y = 6x – 3

y = 2x + 1

y = 2/3x + 6

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 #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?

Possible Answers:

x  y = 3

4x  3y = 4

2x  y = 3

3x + 2y = 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 #5 : 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:

-2

\frac{3}{2}

-\frac{2}{3}

-\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}.

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