1. Section 5-3
Concurrent Lines, Medians, and Altitudes

2. Triangle Medians
A median of a triangle is a line segment drawn from any vertex of the triangle to the midpoint of the opposite side. B D F C A E

3. Centroid Theorem A B C D E F G
The medians of a triangle are concurrent at a point (called the centroid) that is two thirds the distance from each vertex to the midpoint of the opposite side.

4. Triangle Altitudes
An altitude of a triangle is a line segment drawn from any vertex of the triangle to the opposite side, extended if necessary, and perpendicular to that side. A B C E A B C E

5. A B C E F
Why?

6. Triangle Altitude Theorem
The lines that contain the altitudes of a triangle are concurrent (at a point called the orthocenter). A B C E

7. Triangle Perpendicular Bisectors Theorem
The perpendicular bisectors of a triangle are concurrent at a point (called the circumcenter) that is equidistant from the vertices. S Y X C Q R Z
The circle is circumscribed about the triangle.

8. Triangle Angle Bisectors Theorem
The bisectors of the angles of a triangle are concurrent at a point (called the incenter) that is equidistant from the sides. T Y X I U V Z
The circle is inscribed in the triangle.

9. M is the centroid of triangle WOR. WM=16. Find WX.
Application W
M is the centroid of triangle WOR. WM=16. Find WX. Y
WX=24 Z M R O X

10. In triangle TUV, Y is the centroid. YW=9. Find TY and TW.
Application
In triangle TUV, Y is the centroid. YW=9. Find TY and TW. U
TY=18 W
TW=27 X Y V T Z

11. Is KX a median, altitude, neither, or both?
Application K
both L M X

12. Application
Find the center of the circle you can circumscribe about the triangle with vertices:
A (-4, 5); B (-2, 5); C (-2, -2)
Hint: sketch triangle; then think about the perpendicular bisectors passing through the midpoints of the sides!
(-3, 1.5)

13. Application
Find the center of the circle you can circumscribe about the triangle with vertices:
X (1, 1); Y (1, 7); Z (5, 1)
(3, 4)

14. Try these constructions:
1: Circumscribe a circle about a triangle
Draw a large triangle.
Construct the perpendicular bisectors of any two sides. The point they meet is the circumcenter.
The radius is from the circumcenter to one of the vertices. Draw a circle using this radius and it should pass through all three vertices. S Y X C Q R Z

15. Try these constructions:
2: Construct a circle inside a triangle
Draw a large triangle.
Construct the angle bisectors for two of the angles. The point they intersect is called the incenter.
Drop a perpendicular from the incenter to one of the sides. This is your radius.
Draw a circle using this radius and it should touch each side of the triangle. T Y X I U V Z