1. Perpendicular Bisectors and Altitudes of Triangles

2. Perpendicular Bisector
Perpendicular Bisector- A segment, line, ray, or plane that is __________ to another segment at it’s __________.
Equidistant- A point is equidistant from two figures if the point is the same _________ from each figure.

3. Perpendicular Bisector Theorem
In a plane, if a point is on the perpendicular
bisector of a segment, then it is equidistant
from the endpoints of the segment. C
Ex. If CD AB
And
Then A D B T

4. Converse of Perpendicular Bisector Theorem
In a plane, if a point is ________ from the
endpoints of a segment then it is on the
__________________ of the segment. C
A B

5. Perpendicular Bisectors of a Triangle
A line, ray, or segment that is perpendicular to a side of the triangle at the midpoint of the side.

6. Point of Concurrency
Circumcenter- the point of concurrency for the 3 perpendicular bisectors of a triangle.
Right Acute Obtuse
Circumcenter Circumcenter Circumcenter
lies on lies inside lies outside
hypotenuse triangle triangle

7. Circumcenter
It is the same distance from each vertex of the triangle to the circumcenter.
Ex A G
B C

8. Example 1
NO is the perpendicular bisector of LM. If OM = 4 and LN = 6, find LO and MN. L N M O

9. Example 2 2. NO is the perpendicular bisector of
LM. If MN = 6x + 18 and LN = 8x + 6,
find LN and MN. L N M O

10. Example 3 3. The perpendicular bisectors of LMN meet at K. Find LK L N7
4.2

11. Altitudes of Triangles
Altitude- the perpendicular segment from
a vertex of the triangle to the opposite
side or to a line that contains the opposite
side of a triangle A
AD is an altitude
of triangle ABC.
B D C

12. Point of Concurrency
Orthocenter- the point of concurrency for the 3 altitudes of a triangle.
Right Acute Obtuse
Orthocenter Orthocenter Orthocenter
lies on lies inside lies outside
hypotenuse triangle triangle

13. There is absolutely nothing special about the Orthocenter!!!

14. Check
What are the two differences between the perpendicular bisector and the altitude?

15. Example 4
Is BD a perpendicular bisector of ABC? Is BD an altitude of ABC? B D A C B A D C B A C D

16. Example 5
Find the orthocenter of the triangle.
(3, 7)
(8, 3)
(0, 0)