1.Will the following side lengths make a triangle? A. 2, 4, 5 B. 4, 3, 1 2. Find the range of the third side of the triangle: A. 1, 3, x B. 4, 8, x

1.In Triangle DOG, side DO = 25, side OG = 15 and side DG = 30. List the angles in descending order: 2. In Triangle CAT, <C is 60 o, <T is 25 o.List the sides in ascending order:

Angle Bisector

The intersection of the angle bisectors is called the INCENTER. How many angle bisectors does a triangle have?

Point of Concurrency of the Angle Bisectors Always INSIDE the triangle! Equidistant from the SIDES of a triangle

B A C D 12 EX:1

M K L N EX:2 1 2

X W ZY 1 2 EX:3

G F H I EX:4

Perpendicular Bisector

Tell whether each red segment is an perpendicular bisector of the triangle.

The intersection of the perpendicular bisector is called the CIRCUMCENTER. How many perpendicular bisectors does a triangle have?

Point of Concurrency of the Perpendicular Bisectors Can be inside, outside, or on the triangle. Equidistant from the VERTICES of a triangle

Median

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

AE B C D If CD = 2x + 5, BD = 4x – 1, SOLVE FOR X. BD = CD 4x - 1 = 2x + 5 2x = 6 x = 3

The intersection of the medians is called the CENTRIOD. How many medians does a triangle have?

When 3 or more lines (or rays or segments) intersect in the same point, they are called concurrent lines (or rays or segments). The point of intersection of lines is called the point of concurrency.

Point of Concurrency of the Medians Always INSIDE the triangle.

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.

A B F X E C D

A B F X E C D

In  ABC, AN, BP, and CM are medians. A B M P E C N If EM = 3, find EC. EC = 2(3) Ex: 1 EC = 6

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

In  ABC, AN, BP, and CM are medians. A B M P E C N If CM = 3x + 6, and CE = x + 12, what is x? CM = CE Ex: 3 x = 8 (3x + 6) = (x + 12) 2(3x + 6) = 3(x + 12) 6x + 12 = 3x x = 24

A E C B D G 1. If CD = 3.25, what is BC? 2. Find AG if DG = If CG = 7, find CE?

Altitude

Tell whether each red segment is an altitude of the triangle. The altitude is the “true height” of the triangle.

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

Point of Concurrency of the Altitudes Can be inside, outside, or on the triangle.

MEDIAN = CENTROID MC

ALTITUDE = ORTHOCENTER AO

PERPENDICULAR BISECTORS = CIRCUMCENTER PBCC(V)

ANGLE BISECTORS = INCENTER ABI(S)

Equilateral Triangle Right Triangle

The orthocenter, circumcenter, and the centroid are COLLINEAR in EVERY triangle!

A E C B D G 1. If CD = 3.25, what is BC? Determine if each of the following is an Altitude, perpendicular bisector, both, or neither?

Pg 335 # 3 – 10, Write the question and show all your work.