Centers of Triangles or Points of Concurrency

Median

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

How many medians does a triangle have?

The medians of a triangle are concurrent. The intersection of the medians is called the CENTRIOD.

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.

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

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

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

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

Angle Bisector

X W ZY 1 2 Example 1

G F H I Example 2 5(x – 1) = 4x + 1 5x – 5 = 4x + 1 x = 6

How many angle bisectors does a triangle have? three The angle bisectors of a triangle are ____________. concurrent The intersection of the angle bisectors is called the ________. Incenter

The incenter is the same distance from the sides of the triangle. Point P is called the __________. Incenter

Example 4 The angle bisectors of triangle ABC meet at point L. What segments are congruent? Find AL and FL. F D E L B C A 8 LF, DL, EL FL = 6 Triangle ADL is a right triangle, so use Pythagorean thm AL 2 = AL 2 = 100 AL = 10 6

Perpendicular Bisector

Example 1: Tell whether each red segment is a perpendicular bisector of the triangle.

Example 2: Find x 3x + 45x - 10

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.

The Circumcenter is equidistant from the vertices of the triangle. PA = PB = PC

6 D B A C P 10 Example 3: The perpendicular bisectors of triangle ABC meet at point P. Find DA. Find BA. Find PC. Use the Pythagorean Theorem to find DP. DA = 6 BA = 12 PC = 10 DP = 10 2 DP = 100 DP 2 = 64 DP = 8

Altitude

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? ALTITUDE NEITHER BOTH PER. BISECTOR