5 3 CONCURRENT LINES MEDIANS AND ALTITUDES PART

5. 3 CONCURRENT LINES, MEDIANS AND ALTITUDES PART B LEQ: How do we construct and use medians and altitudes?

Median of a triangle: a segment whose endpoints are a vertex and the midpoint of the opposite side Altitude of a triangle: the perpendicular segment from a vertex to the line containing the opposite side

Theorem 5 -8 The medians of a triangle are concurrent (intersect) at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. D Centroid DC = 2/3 DJ EC = 2/3 EG FC = 2/3 FH H G The point of intersection is C called the centroid. F J E

Example C is the centroid. 1. ) If CJ = 12, find DC and DJ. DC=2/3 (DJ) and CJ=1/3 (DJ) 2. ) If FC = 15, find CH and FH. D 3. If GE = 30, find GC and CE. H G C F J E

Bubble sheet Centroid: Medians (vertex to midpoint) 2: 3/1: 3 rule Pieces of median are a 2: 1 ratio

Example Is ST a median, an altitude or neither? What is UW? What is VX? What is CA? A S W X V C T U

The point of concurrency of the altitudes is called the orthocenter. The point of concurrency of the medians is called the centroid.

Orthocenter: Altitudes/height (vertex to opposite side & makes right angle) May be inside of, outside of, or on triangle

Let’s practice: Identify each point of concurrency below: a b c d