3.4 Medians, Altitudes & (Slightly) More Complex Proofs

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

Three Medians per Triangle

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

THREE ALTITUDES PER TRIANGLE

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.

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

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

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

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

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