>Angles>What Does Normal Mean in Geometry?

What Does Normal Mean in Geometry?

A normal is a line that makes a $90$ $^{\circ}$ 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 $90$ $^{\circ}$ 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 $90$ $^{\circ}$ 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 $90$ $^{\circ}$ angle.

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 60°, 30° or 15° Angle

Constructing a 90°, 45° or 22.5° Angle

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Perpendicular Bisector

Instructions for Constructing the Perpendicular Bisector of a Line

Normal from the Point P to a Line

Instructions for Constructing a Normal from Point to Line

Normal from the Point $P$ to a Line