1. 3.4 Medians, Altitudes & (Slightly) More Complex Proofs

2. Medians
Definition: A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.
Median
Midpoint

3. Three Medians per Triangle

4. Altitudes
Definition: An altitude of a triangle is a segment drawn from the vertex of a triangle perpendicular to the (line containing) the opposite side.
Can be inside, on a side, or outside the triangle
Acute Triangle:
Altitude inside
Right Triangle:
Altitude is a side
Obtuse Triangle:
Altitude outside

5. THREE ALTITUDES PER TRIANGLE

6. Altitude - Special Segment of Triangle
In a right triangle, two of the altitudes of are the legs of the triangle. B A D F B A D F I K
In an obtuse triangle, two of the altitudes are outside of the triangle.

7. Two Points Determine a Line
Postulate:
The Line Postulate
Two Points Determine a Line
(or line segment)
Abbreviated: LP

8. Auxiliary Lines
Additional lines needed, NOT drawn in the given diagram. A
Given:
Prove :
Easily proven if we had triangles, but we don’t. B
Or do we?
Any two points determine a line (or segment) D C
Now triangles are congruent by SSS and angles are congruent by CPCTC

9. Write the proof now in 2 columns
Given:
Prove : A
When you introduce a line segment into a diagram write:
Statement
Reason
Let
Line Postulate
(LP) B D C

10. Identifying Medians and Altitudes
Is KX a median, an altitude, neither, or both?
Both

11. Homework Help