Centers of Triangles or Points of Concurrency. Medians. Median. vertex to midpoint. Example 1. M. D. P. C. What is NC if NP = 18?. MC bisects NP…so 18/2. 9. N. If DP = 7.5, find MP. 15. 7.5 + 7.5 =. How many medians does a triangle have?. Three – one from each vertex.
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vertex to midpoint
What is NC if NP = 18?
MC bisects NP…so 18/2
If DP = 7.5, find MP.
7.5 + 7.5 =
Three – one from each vertex
The medians of a triangle are concurrent.
The intersection of the medians is called the CENTRIOD.
They meet in a single point.
The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint.
If EM = 3, find EC.
EC = 2(3)
EC = 6
If EN = 12, find AN.
AE = 2(12)=24
AN = AE + EN
AN = 24 + 12
AN = 36
Example 4In ABC, AN, BP, and CM are medians.
If EM = 3x + 4 and CE = 8x, what is x?
x = 4
Example 5In ABC, AN, BP, and CM are medians.
If CM = 24 what is CE?
CE = 2/3CM
CE = 2/3(24)
CE = 16
vertex to side cutting angle in half
5(x – 1) = 4x + 1
5x – 5 = 4x + 1
x = 6
The angle bisectors of a triangle are ____________.
The intersection of the angle bisectors is called the ________.
Point P is called the __________.
The angle bisectors of triangle ABC meet at point L.
LF, DL, EL
Triangle ADL is a right triangle, so use Pythagorean thm
AL2 = 82 + 62
AL2 = 100
AL = 10
FL = 6
Perpendicular Bisector triangle.
midpoint and perpendicular
(don't care about no vertex)
Example 1: Tell whether each red segment is a triangle.perpendicular bisector of the triangle.
Example 2: Find x triangle.
3x + 4
5x - 10
x = 7
The perpendicular bisectors of a triangle are concurrent.
The intersection of the perpendicular bisectors is called the CIRCUMCENTER.
PA = PB = PC
DA = 6
BA = 12
PC = 10
DP2 + 62 = 102
DP2 + 36 = 100
DP2 = 64
DP = 8
Altitude at point P.
vertex to opposite side and perpendicular
Tell whether each red segment is an at point P. altitude of the triangle.
The altitude is the “true height” of the triangle.
How many altitudes does a triangle have? at point P.
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?