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.
Centers of Triangles or Points of Concurrency
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 =
How many medians does a triangle have?
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
If EM = 3x + 4 and CE = 8x, what is x?
x = 4
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
How many angle bisectors does a triangle have?
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
midpoint and perpendicular
(don't care about no vertex)
Example 1: Tell whether each red segment is a perpendicular bisector of the triangle.
Example 2: Find x
3x + 4
5x - 10
x = 7
How many perpendicular bisectors does a triangle have?
The perpendicular bisectors of a triangle are concurrent.
The intersection of the perpendicular bisectors is called the CIRCUMCENTER.
PA = PB = PC
Example 3: The perpendicular bisectors of triangle ABC meet at point P.
DA = 6
BA = 12
PC = 10
DP2 + 62 = 102
DP2 + 36 = 100
DP2 = 64
DP = 8
vertex to opposite side and perpendicular
Tell whether each red segment is an altitude of the triangle.
The altitude is the “true height” of the triangle.
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?