Centers of Triangles or Points of Concurrency

Centers of Triangles or Points of Concurrency Geometry October 28, 2013

OBJECTIVE You will learn how to construct perpendicular bisectors, angle bisectors, medians and altitudes of triangles constructed

Today’s Agenda • Triangle Segments • Median • Altitude • Perpendicular Bisector • Angle Bisector • Triangle Centers • Centroid • Orthocenter • Circumcenter • Incenter

VOCABULARY • A median of a triangle is a segment that connects a vertex to the midpoint of the opposite side. • The altitude of a triangle is the perpendicular distance from one of its bases to the opposite vertex. In other words, the altitude is a segment that is perpendicular to one side and reaches the point across from that side. • A perpendicular bisector of a line segment isa) perpendicular to it, and b) bisects it. • An angle bisector of a triangle is a segment that divides one of its angles into two congruent pieces. The segment connects to the opposite side

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

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

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

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 E AN = 24 + 12 B AN = 36 M A

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

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

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

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

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

Altitude Altitude 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. YES NO YES

How many altitudes does a triangle have? Three The altitudes of a triangle are concurrent. The intersection of the altitudes is called the ORTHOCENTER.

Perpendicular Bisector Perpendicular Bisector midpoint and perpendicular (don't care about no 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

How many perpendicular bisectors does a triangle have? Three 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 BA = 12 • Find BA. • Find PC. PC = 10 • Use the Pythagorean Theorem to find DP. B 6 DP2 + 62 = 102 DP2 + 36 = 100 DP2 = 64 DP = 8 10 D P A C

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

Sum It Up bisector bisector which is… Figure concurrent at.. circumcenter equidistant from vertices incenter equidistant from sides median centroid 2/3 distance from vertices to midpoint altitude orthocenter ---------------------

Centers of Triangles or Points of Concurrency