Centers of Triangles or Points of Concurrency Median

Example 1 M D C N P What is NC if NP = 18?

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

Theorem The length of the segment from the vertex to the centroid is twice

Example 2 In ABC, AN, BP, and CM are medians. If EM = 3,

Example 3 In ABC, AN, BP, and CM are medians. If EN = 12,

Example 4 In ABC, AN, BP, and CM are medians. If EM = 3

Example 5 In ABC, AN, BP, and CM are medians. If CM = 24

How many angle bisectors does a triangle have? three The angle bisectors of a

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

Example 4 The angle bisectors of triangle ABC meet at point L. • What

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

How many perpendicular bisectors does a triangle have? The perpendicular bisectors of a triangle

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

Example 3: The perpendicular bisectors of triangle ABC meet at point P. • Find

Tell whether each red segment is an altitude of the triangle. The altitude is

How many altitudes does a triangle have? The altitudes of a triangle are concurrent.

Tell if the red segment is an altitude, perpendicular bisector, both, or neither? NEITHER

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Centers of Triangles or Points of Concurrency

Median

Example 1 M D C N P What is NC if NP = 18? MC bisects NP…so 18/2 9 If DP = 7. 5, find MP. 7. 5 + 7. 5 = 15

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.

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

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

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

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

Angle Bisector

Example 1 W X 1 Z 2 Y

Example 2 F G I 5(x – 1) = 4 x + 1 H 5 x – 5 = 4 x + 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? LF, DL, EL • Find AL and FL. Triangle ADL is a A right triangle, so use FL = 6 Pythagorean thm 8 D AL 2 = 82 + 62 AL 2 = 100 F 6 C AL = 10 L E B

Perpendicular Bisector

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

Example 2: Find x 3 x + 4 5 x - 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

Example 3: The perpendicular bisectors of triangle ABC meet at point P. • Find DA. DA = 6 • Find BA. BA = 12 • Find PC. PC = 10 • Use the Pythagorean Theorem B to find DP. 6 DP 2 + 62 = 102 DP 2 D + 36 = 100 DP 2 = 64 DP = 8 10 P A C

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? NEITHER ALTITUDE BOTH PER. BISECTOR