Two intersecting lines are perpendicular if they form a right angle (an angle that measures ).
For example, the two lines above are perpendicular because the angle is a right angle.
The opposite is also true: if two lines are perpendicular, then they meet at a right angle.
As you may have noticed already, when two lines are perpendicular, they don't form only one right angle. All the four angles formed by the intersection are right angles!
A segment bisector is any segment, line, or ray that splits another segment into two congruent parts.
For example, in the diagram below, is a bisector of the segment
A perpendicular bisector is any segment, line, or ray that bisects another segment and is perpendicular to that segment.
For example, in the diagram below, is a perpendicular bisector of the segment
Which of the following diagrams represents a perpendicular bisector of
A perpendicular bisector of :
must split in half, and
must be perpendicular to .
Among the given options, only diagram IV shows a perpendicular bisector to the line segment
Notice that in diagrams I, II and III we have bisectors but they are not perpendicular to
In the diagram above, $\overset{\longleftrightarrow}{UV}$ is a perpendicular bisector of $\overline{ST}.$ If $SO=8,$ then $ST =$
|
a
|
$2$ |
|
b
|
$16$ |
|
c
|
Not enough information |
|
d
|
$4$ |
|
e
|
$8$ |
Which of the diagrams above represents a perpendicular bisector of $\overline{AB}?$
|
a
|
I and IV only |
|
b
|
II and III only |
|
c
|
III only |
|
d
|
I only |
|
e
|
IV only |
The perpendicular bisector theorem states that:
In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
For example, in the diagram below, we see that is the perpendicular bisector of
Consequently, we can conclude that the point is equidistant from the endpoints of So we have
Solve for in the figure below given that is the perpendicular bisector of
By the perpendicular bisector theorem, a point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. Hence, we have:
Solve for $x$ in the figure above given that $\overset{\longleftrightarrow}{PQ}$ is the perpendicular bisector of $\overline{MN}.$
|
a
|
$7$ |
|
b
|
$5$ |
|
c
|
$6$ |
|
d
|
$3$ |
|
e
|
$2$ |
Solve for $x$ in the figure above given that $\overset{\longleftrightarrow}{CD}$ is the perpendicular bisector of $\overline{AB}.$
|
a
|
$9$ |
|
b
|
$12$ |
|
c
|
$3$ |
|
d
|
$2$ |
|
e
|
$18$ |
The converse of the perpendicular bisector theorem states that:
In a plane, if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
For example, in the diagram below, we see that is perpendicular to and the point is equidistant from the endpoints of
Therefore, we can conclude that is the perpendicular bisector of In particular, we conclude that
Solve for in the figure below given that and
By the converse of the perpendicular bisector theorem, a point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.
Here, the points and are both equidistant from the endpoints of Therefore, and both lie on the perpendicular bisector of
Since only one line can be drawn through two given points, the line is the perpendicular bisector of
In particular, we have that We can use this equation to solve for
Solve for $x$ in the figure above given that $AD=BD$ and $AE=BE.$
|
a
|
$1$ |
|
b
|
$4$ |
|
c
|
$3$ |
|
d
|
$6$ |
|
e
|
$9$ |
Solve for $x$ in the figure above given that $KN=LN$ and $KP=LP.$
|
a
|
$5$ |
|
b
|
$4$ |
|
c
|
$1$ |
|
d
|
$8$ |
|
e
|
$2$ |
The most important property of perpendicular lines is the following.
Given a line and a point , there is one and only one line that passes through and is perpendicular to .
Other nice properties include the following:
If two distinct lines are perpendicular to the same (third) line, then these two lines are parallel. For example, in the figure below, both lines and are perpendicular to the line . As a result, .
If a line is perpendicular to one of two parallel lines, then that line is also perpendicular to the second parallel line.
Symbol
Finally, if a line is perpendicular to a line , we can denote it without words using the symbol (perpendicular), as follows:
This reads as " is perpendicular to ."