Centers of Triangles or Points of Concurrency Prepared for Ms. Pullo’s Geometry Classes

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 side cutting angle in half Angle Bisector Angle Bisector vertex to side cutting angle in half

Example 1 W X 1 2 Z Y

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

three concurrent Incenter 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

Triangle ADL is a right triangle, so use Pythagorean thm Example 4 The angle bisectors of triangle ABC meet at point L. What segments are congruent? Find AL and FL. LF, DL, EL Triangle ADL is a right triangle, so use Pythagorean thm AL2 = 82 + 62 AL2 = 100 AL = 10 F D E L B C A 8 FL = 6 6

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.

Perpendicular Bisector midpoint and perpendicular (MAY not come from vertex)

Example 1: Tell whether each red segment is a perpendicular bisector of the triangle. NO NO YES

Example 2: Find x 3x + 4 5x - 10 x = 7

Three 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 to find DP. B DP2 + 62 = 102 DP2 + 36 = 100 DP2 = 64 DP = 8 6 10 D P A C

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

IN A NUT SHELL Median – Centroid Angle Bisector – Incenter Altitude – Orthocenter Perpendicular Bisector - Circumcenter Angle Bisector: The Incentor is equidistance to the sides Perpendicular Bisector – the Circumcenter is equidistance to the vertex

The End Study!!!