1. Relationships in Triangles
Bisectors, Medians, and Altitudes
Section 6.1 – 6.3
Students Should Begin Taking Notes At Screen 4!!

2. Objectives of this lesson
To identify and use perpendicular
bisectors & angle bisectors in triangles
To identify and use medians & altitudes in triangles

3. Vocabulary Perpendicular Bisectors Angle Bisectors Medians Altitudes
Points of Concurrency

4. Perpendicular bisector
STUDENTS SHOULD BEGIN TAKING NOTES HERE!
Perpendicular bisector
A line segment or a ray that passes through the midpoint of a side of a triangle & is ⊥ to that side.
In the picture to the right, the red line segment is the ⊥ bisector

5. Perpendicular Bisector (con’t)
For every triangle there are 3 perpendicular bisectors
The 3 perpendicular bisectors intersect in a common
point named the circumcenter.
In the picture to the right
point K is the circumcenter.

6. Perpendicular Bisector (con’t)
Any point on the perpendicular bisector of a segment is equidistant
from the endpoints of the segment
Any point equidistant from the endpoints of a segment lies on the
perpendicular bisector of the segment

7. Angle Bisector
A line, line segment or ray that bisects an interior angle of a triangle
In the picture to the right, the red line segment is the angle bisector. The  arc marks show the 2  angles that were formed when the angle bisector bisected the original angle.

8. Angle Bisector (con’t)
For every triangle there are 3 angle bisectors.
The 3 angle bisectors intersect in a common point named the incenter
In the picture to the right, point I is the incenter.

9. Angle Bisector (con’t)
Any point on the angle bisector is equidistant from the sides of the angle.
Any point equidistant from the sides of an angle lies on the angle bisector.

10. Median A line segment whose endpoints are a vertex of a
triangle and the midpoint of the side opposite the
vertex.
In the picture to the right, the blue line segment is the median.

11. Median (con’t) For every triangle there are 3 medians
The 3 medians intersect in a common point named the centroid
In the picture to the right, point O is the centroid.

12. Altitudes
A line segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side.
In the picture above, ∆ABC is an obtuse triangle & ∠ACB is the obtuse angle. BH is an altitude.

13. Altitudes (con’t) For every triangle there are 3 altitudes
The 3 altitudes intersect in a common point called
the orthocenter.
In the picture to the right, point H is the orthocenter.

14. Points of Concurrency Concurrent Lines
3 or more lines that intersect at a common point
Point of Concurrency
The point of intersection when 3 or more lines intersect.
Type of Line Segments Point of Concurrency
Perpendicular Bisectors Circumcenter
Angle Bisectors Incenter
Median Centroid
Altitude Orthocenter

15. Points of Concurrency (con’t)
Facts to remember:
The circumcenter of a triangle is equidistant from the
vertices of the triangle.
Any point on the angle bisector is equidistant from the sides of the angle (Converse of #3)
Any point equidistant from the sides of an angle lies on
the angle bisector. (Converse of #2)
The incenter of a triangle is equidistant from each side of the triangle.
The distance from a vertex of a triangle to the centroid is 2/3 of the median’s entire length. The length from the centroid to the midpoint is 1/3 of the length of the median.

16. Points of Concurrency (con’t)

17. Facts To Remember & MEMORIZE!
1. Perpendicular Bisectors
2. Angle Bisectors
3. Medians
4. Altitudes
1. …form right angles AND  lines segments
2. …form 2  angles
3. …form 2  line segments
4. … form right angles

18. The End
(Finally!)