Geometry / Unit 6: Relationships in Triangles
6.2 Perpendicular and Angle Bisectors
Notes
Bisectors give equal distances.
- Perpendicular Bisector Theorem: a point on the perpendicular bisector of a segment is equidistant from the segment's endpoints. The converse is also true.
- Angle Bisector Theorem: a point on the bisector of an angle is equidistant from the two sides of the angle (distance measured perpendicular to the sides). The converse is also true.
- Example: $P$ is on the perpendicular bisector of $\overline{AB}$, $PA = 3x$ and $PB = x + 8$. Then $3x = x + 8$, so $x = 4$.
Practice
$P$ lies on the perpendicular bisector of $\overline{JK}$. $PJ = 4x - 3$ and $PK = 2x + 9$. Find $PJ$.
$4x - 3 = 2x + 9$ gives $x = 6$, so $PJ = 21$.
Point $D$ is on the bisector of $\angle ABC$. The perpendicular distance from $D$ to $\overrightarrow{BA}$ is 7. What is the perpendicular distance from $D$ to $\overrightarrow{BC}$?
Points on an angle bisector are equidistant from the sides.
Point $X$ is 5 units from $A$ and 5 units from $B$. What can you conclude?
By the converse of the Perpendicular Bisector Theorem. $X$ need not be on $\overline{AB}$.
Ready for more? Take the Unit 6 test.