Constructing  perpendiculars

Constructing perpendiculars to a line through a specific given point using only a straightedge and a compass is what I will show you here. There are two cases that you may encounter.

Case #1: You are given a line and a point on the line.

Constructing a perpendicular when point is on the line


Case #2: You are given a line and a point not on the line.

Constructing a perpendicular when point is not on line

We will demonstrate both cases.

Constructing perpendiculars: Let us start with case #1.

Step #1

Start with the following line and point P

Constructing a perpendicular when point is on the line


Open your compass to any radius or distance. Put the needle of the compass at point P and draw arcs on both sides of point P. Label the points where the arcs intersect the line C and D.

Constructing a perpendicular when point is on line: step 1

Step #2

Open your compass to a radius or distance greater than the distance you chose to draw the arcs.

After you have increased your compass opening, place the needle at C and draw and arc.

Then, still using the same compass opening place the needle at D and draw another arc.

The two arcs should intersect as shown below. Call the point the two arcs intersect E.

Constructing a perpendicular when point is on line: step 2


Step #3

Draw a line between P and E and you are done!

Constructing a perpendicular when point is on line: step 3

Constructing perpendiculars: case #2.


Case #2: You are given a line and a point not on the line.

Start with the following line and point P not on the line.

Constructing a perpendicular when point is not on the line

Step #1

Open your compass to a radius or distance greater than the distance from P to the line.

The distance you see below in red is the shortest distance from P to the line. That is what I mean by distance from P to the line; It is the shortest distance from P to the line which is usually done by drawing a straight line down.

Constructing a perpendicular when point is not on the line: step 1

Open you compass to a distance greater than that.

Step #2

After opening your compass to a distance greater than the shortest distance from P to the line, place the needle of the compass at P and draw arcs that intersect the line in two places.

Constructing a perpendicular when point is not on the line: step 2

Call the points the arc intersect the line A and B

Step #3

Keeping the opening of the compass the same, place the needle at A and draw and arc.

Then, place the needle at B and draw an arc. Call the point the two arc intersect E.

Constructing a perpendicular when point is not on the line: step 3

Step #4

Draw a line between point P and point E and you are done!

Constructing a perpendicular when point is not on the line: step 4

Here we go! Make sure you understand each step outlined above and follow them carefully. If you do this, you will have no problems whatsoever constructing perpendiculars.

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