How to Construct a Perpendicular Line to a Given Line Through Point on the Line

Co-authored by Kevin Wang

Last Updated: June 22, 2023 Fact Checked

Perpendicular lines are lines that form right angles. These types of lines are used often in geometry, so it is important to know how to construct them. The simplest way to construct a perpendicular line is to use a protractor. However, you will often be asked to construct a perpendicular line using only a compass and straightedge.

Method 1

Method 1 of 2:

Using a Protractor

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1
Line up the given line and the protractor. Set the protractor’s origin hole over the line's given point. Align the protractor so that its base line sits exactly over the given line.^{[1]
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- The origin hole is the small hole at the bottom-center of the protractor.
- The base line is the line on the bottom of the protractor marking 180/0 degrees.
This is the line measurement at the very top of the protractor. Two perpendicular lines form a 90-degree, or right, angle. Thus, to draw a perpendicular line, you need to draw a 90-degree angle.^{[2]
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Use the straightedge of the protractor to ensure that the line is straight. This line is perpendicular to the original line, through the given point on the line.

Method 2

Method 2 of 2:

Using a Compass and Straight Edge

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1
Draw an arc on either side of the given point. To do this, place the compass tip on the given point on the line. Then, swing the compass, drawing two arcs on both sides of the given point. The arcs should intersect the line.^{[3]
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Research source} Mark and label the points where the arcs intersect the line.
- You can set the compass to any width for this step.
- For example, you might be given point A on a line. Use the compass to draw point P to the left, and point Q to the right.
2
Increase the width of the compass. Place the compass tip on the new point to the left of the original point. Stretch the compass so that it reaches about halfway between the original point, and the new point on the right.^{[4]
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- For example, place the compass tip on point P, and stretch the compass to a point about halfway between points A and Q.
3
Draw an arc above the line. You could also draw the arc below the line. Keeping the compass tip on the left point, swing the compass, drawing an arc. Make sure the arc crosses over the original point.^{[5]
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- For example, keep the compass tip on point P. Draw an arc above the line.
4
Draw a second arc, intersecting the first. Do not change the width of the compass. Set the compass tip on the right point. Swing the compass, drawing an arc above the line that intersects the first arc.^{[6]
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Research source} Mark this intersection with another point.
- For example, place the compass tip on point Q. Draw an arc that intersects the first arc at point T.
5
Connect the given point to the point where the arcs intersect. Use a straightedge to ensure the line is straight. The line you draw is perpendicular to the first line, through the given point on the line.^{[7]
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- For example, draw a line connecting points A and T.

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Question

How do I find the equation of the line?

Kevin Wang
Math Tutor
Kevin Wang is a Math Tutor based in New York, New York. Kevin has been tutoring math for over 10 years, and specializes in K-12 math topics and standardized tests, such as SAT and ACT. Kevin has an economics background and a career in both finance and marketing analytics. His interest in tutoring goes back even longer than his career. He discovered tutoring at the start of his university career and enjoys it as a way to stay sharp with fundamental skills and remain up to date with trends in our education system. Kevin received a BS in Economics from Duke University.

Kevin Wang

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Expert Answer

To find a line perpendicular to another line, two things must be done: first, flip the slope of the other line upside down and change its sign. For example, if it is 2, the slope of a perpendicular line would be -1/2. Second, pick a point that the new line goes through. Once you have the slope and have a point, you can now use the point formula to find the equation of the perpendicular line.

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Can we draw the arc both above/below the line?

Donagan

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Yes. With either of the methods shown above, you may (but don't have to) draw similar arcs on both sides of the given line. An arc (or arcs) on just one side of the line is sufficient, but drawing them on both sides is a good way to verify the accuracy of the perpendicular line.

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Question

How can I divide a straight line into 6 equal parts?

Donagan

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Assuming no measuring tool or protractor, this can be done with a compass and straightedge. Label the line AB. From A draw another straight line at any acute angle from AB. Mark off six equal segments on this new line. Label the segments AC, CD, DE, EF, FG and GH. Draw a line from H to B. From line AH draw five lines to AB that are each parallel to HB. (Use a compass or protractor to make these lines parallel to HB.) These parallel lines will intersect AB so as to divide AB into six equal parts.

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