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Digital SAT · Solve it in Desmos

How to Solve Parallel and Perpendicular Lines on Desmos (Digital SAT)

A graph confirms the relationship and that the line hits the point, which catches a wrong slope. Work the slope rule by hand first, then let the picture verify it.

Perfect1600 Content Team·Uses the same Desmos calculator that is built into the Digital SAT

Full method for parallel and perpendicular lines

The Desmos steps

  1. 1

    Graph both lines

    Type the original and your new line into Desmos.

  2. 2

    Check the relationship

    Parallel lines never touch; perpendicular lines meet at a right angle.

  3. 3

    Confirm the point

    See that your line runs through the required point.

Open a real question in the same Desmos calculator you get on Perfect1600, with the setup already typed in:

Worked in Desmos

Easy example
In the xyxy-plane, line mm has equation y=2x+5y=2x+5. What is the slope of a line perpendicular to line mm?
A
22
−12-\frac{1}{2}
C
12\frac{1}{2}
D
−2-2

A: Incorrect. This keeps the slope, which gives a parallel line.

B: Correct. The perpendicular slope is the negative reciprocal of 2, which is −12-\frac{1}{2}.

C: Incorrect. This is the reciprocal without the sign flip.

D: Incorrect. This negates the slope but does not take the reciprocal.

Explanation

Perpendicular slope is the negative reciprocal of 2: -1/2.

Medium example
Line ℓ\ell passes through the points (1,2)(1, 2) and (5,10)(5, 10). Which equation defines a line perpendicular to ℓ\ell?
A
y=−2x+7y = -2x + 7
y=−12x+7y = -\frac{1}{2}x + 7
C
y=12x+7y = \frac{1}{2}x + 7
D
y=2x+7y = 2x + 7

A: Incorrect. This negates the slope of ℓ\ell but does not take the reciprocal.

B: Correct. The slope of ℓ\ell is 2, so the perpendicular slope is −12-\frac{1}{2}.

C: Incorrect. This is the reciprocal with the wrong sign.

D: Incorrect. This is the slope of ℓ\ell itself, which is parallel, not perpendicular.

Explanation

Slope of ℓ\ell is 10−25−1=2\frac{10 - 2}{5 - 1} = 2. The perpendicular slope is −12-\frac{1}{2}.

Hard example
In the xyxy-plane, a line passes through the point (0,5)(0,5) and is perpendicular to the line y=25x−1y=\frac{2}{5}x-1. Which equation represents this line?
A
y=25x+5y=\frac{2}{5}x+5
B
y=52x+5y=\frac{5}{2}x+5
y=−52x+5y=-\frac{5}{2}x+5
D
y=−52x−1y=-\frac{5}{2}x-1

A: Incorrect. This uses the parallel slope, not the perpendicular slope.

B: Incorrect. This is the reciprocal without the sign flip.

C: Correct. The perpendicular slope is −52-\frac{5}{2}, and passing through (0,5)(0,5) gives intercept 5, so y=−52x+5y=-\frac{5}{2}x+5.

D: Incorrect. This line passes through (0,−1)(0,-1), not (0,5)(0,5).

Explanation

Perpendicular slope -5/2 through (0,5): y = -5/2 x + 5.

Try it with the calculator

Question 1easy

Line ℓ\ell in the xy-plane has the equation y=3x+2y = 3x + 2. A line parallel to ℓ\ell passes through the origin. Which equation could define the parallel line?

Question 2easy
Graph
-8-6-4-22468-10-5510Oxy

A line in the xy-plane is perpendicular to the line y=4x−1y = 4x - 1. What is the slope of this perpendicular line?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

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Desmos questions

Can Desmos solve parallel and perpendicular lines questions on the SAT?

A graph confirms the relationship and that the line hits the point, which catches a wrong slope. Work the slope rule by hand first, then let the picture verify it.

Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

Master parallel and perpendicular lines, calculator and all

Every Digital SAT question type comes with the method, the Desmos shortcut, and free practice with instant feedback.

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