How to Draw Perpendicular Lines in Geometry

Last Updated: October 16, 2016

Drawing perpendicular lines is quite easy to do, and you have probably accidentally done it before. All you need is a piece of paper and a pencil. If you don't know what this is, I will teach you how to do it.

Method 1

Method 1 of 4:

Bisection Construction

This construction is explained in more detail in Bisect a Line With a Compass and Straightedge.

1
Draw one of the lines and mark two points on it.
2
Set a compass to at least half the distance between the two points.

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3
Use the compass to draw a circle centered around each point. The circles should intersect in two points on opposite sides of the line.
4
Draw a line through the two points of intersection. This is perpendicular to the original line. It also passes exactly halfway between the two points used as the centers.

Method 2

Method 2 of 4:

Pythagorean

Another variation of this method is given in Form a Perfect Right Angle Using Just String.

1
Draw a line of length 5 (in any units). Either use a ruler, or use a compass to mark off the same distance 5 times as shown.
2
Draw a circle of radius 3 around one of the endpoints.
3
Draw a circle of radius 4 around the other endpoint.
4
Draw a line from the intersection of these circles to each of the endpoints. This creates a triangle with side lengths of 3, 4, and 5. The Pythagorean Theorem states that the angle opposite the side of length 5 is a right angle.

Method 3

Method 3 of 4:

Thales' Method

1
Draw a line with a straightedge.
2
Draw a circle whose center lies on the line. Note the two endpoints of the diameter.
3
Pick any point on the circle and draw the lines connecting it to each endpoint of the diameter. These lines are perpendicular.

Method 4

Method 4 of 4:

Coordinate Geometry

1
Take the equation of a line and write it in the standard form, Ax + By = C. Note the coefficients A and B (C is unimportant unless you want to ensure that the line passes through some particular point).
2
Write the equation of another line, Bx - Ay = D. Note that the coefficients of x and y have switched and one of them changed sign. D can be chosen arbitrarily as it only affects the point of intersection, not the angle.
3
Graph both lines. They will be perpendicular.

Community Q&A

Question

What is the simplest way to draw perpendicular lines?

Donagan

Top Answerer

The simplest way is to use a protractor or a right-angle template such as a drafting triangle.
Question

What is the easiest way to prove that a quadrilateral is cyclic?

Donagan

Top Answerer

A cyclic quadrilateral is a rectangle (or square). The easiest way to prove a quadrilateral is a rectangle is to prove that the opposite sides are equal. You could also prove that the opposite angles are equal or that the diagonals are equal.
Question

How can we draw a perpendicular on a straight line where no point is given?

Community Answer

If you're just trying to draw a line perpendicular to a given line with no particular point of intersection in mind, use Method 1 above.