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What Does Normal Mean in Geometry?

A normal is a line that makes a 90° angle with another line, meaning that it’s a line that makes a right angle with another line. This will almost always come up in a construction, so make sure you remember this.

Perpendicular Bisector

A perpendicular bisector is a line that makes a 90° angle in the exact middle of another line segment. In other words, the perpendicular bisector is a normal that splits a line segment into two equal parts. You construct a perpendicular bisector in the following way:

Rule

Instructions for Constructing the Perpendicular Bisector of a Line

1.
You have a line segment l with two points A and B.
2.
Put the point of your draft compass on A and make two small arcs roughly above and below the midpoint between A and B. In other words, one arc above the line l and one arc below the line l.
3.
Without changing the distance between the legs of the draft compass, put the point of the draft compass on B and do the same thing. Make sure the little arcs intersect to make two crosses—one above l and one below l.
4.
Draw a straight line between the two crosses you just made with your draft compass.
5.
This line is the locus of points with equal distance to A and B. The line makes a 90° angle with l. We call this line the perpendicular bisector.

The only difference between the two normals—the perpendicular bisector and a normal from a point to a line—is the procedure for constructing them. Both yield the same result: Two lines forming a 90° angle.

Perpendicular bisector and normal from point to line

Perpendicular bisector and normal from point to line

Normal from the Point P to a Line

Rule

Instructions for Constructing a Normal from Point to Line

1.
Draw the line l and the point P outside the line.
2.
Put the point of the draft compass on P and make an arc that intersects the line l in two places. Call the intersections A and B.
3.
Move the point of the draft compass to A and make an arc on the opposite side of l to P.
4.
Do the same from B without changing the distance between the two legs of the draft compass, and make sure the two arcs intersect.
5.
Draw a line from P to the new intersection.
6.
You have now drawn the normal from point P to the line.

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Constructing a 90°, 45° or 22.5° Angle