Medians Median vertex to midpoint

Example 1 M D P C N What is NC if NP = 18? MC bisects NP…so 18/2 9 If DP = 7.5, find MP. 15 7.5 + 7.5 =

Three – one from each vertex How many medians does a triangle have? Three – one from each vertex

They meet in a single point. The medians of a triangle are concurrent. The intersection of the medians is called the CENTRIOD. They meet in a single point.

Theorem The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint. 2x x

In ABC, AN, BP, and CM are medians. Example 2 In ABC, AN, BP, and CM are medians. If EM = 3, find EC. C EC = 2(3) N P E EC = 6 B M A

In ABC, AN, BP, and CM are medians. Example 3 In ABC, AN, BP, and CM are medians. If EN = 12, find AN. C AE = 2(12)=24 AN = AE + EN N P AN = 24 + 12 E B AN = 36 M A

In ABC, AN, BP, and CM are medians. Example 4 In ABC, AN, BP, and CM are medians. If EM = 3x + 4 and CE = 8x, what is x? A B M P E C N x = 4

In ABC, AN, BP, and CM are medians. Example 5 In ABC, AN, BP, and CM are medians. If CM = 24 what is CE? A B M P E C N CE = 2/3CM CE = 2/3(24) CE = 16

vertex to opposite side and perpendicular Altitude Altitude vertex to opposite side and perpendicular

The altitude is the “true height” of the triangle. Tell whether each red segment is an altitude of the triangle. The altitude is the “true height” of the triangle. YES NO YES

Three How many altitudes does a triangle have? The altitudes of a triangle are concurrent. The intersection of the altitudes is called the ORTHOCENTER.

Tell if the red segment is an altitude, perpendicular bisector, both, or neither? PER. BISECTOR BOTH