3.1 Special Segments and Centers of Classifications of Triangles: Triangles By Side: 1. Equilateral: A triangle with three I CAN... congruent sides. Define and recognize perpendicular bisectors, angle bisectors, medians, 2. Isosceles: A triangle with at least and altitudes. two congruent sides. 3. Scalene: A triangle with three sides Define and recognize points of having different lengths. (no sides are concurrency. congruent)

Jul 24­9:36 AM Jul 24­9:36 AM

Classifications of Triangles: Special Segments and Centers in Triangles

By angle A Perpendicular Bisector is a segment or line 1. Acute: A triangle with three acute that passes through the midpoint of a side and angles. is perpendicular to that side. 2. Obtuse: A triangle with one obtuse angle. 3. Right: A triangle with one right angle 4. Equiangular: A triangle with three congruent angles

Jul 24­9:36 AM Jul 24­9:36 AM

Point of Concurrency The three perpendicular bisectors of a triangle intersect at a single point. Two lines intersect at a point. The point of When three or more lines intersect at the concurrency of the same point, it is called a "Point of perpendicular Concurrency." bisectors is called the circumcenter.

Jul 24­9:36 AM Jul 24­9:36 AM 1 Circumcenter Properties An angle bisector is a segment that divides 1. The circumcenter is an angle into two congruent angles. the center of the circumscribed circle.

BD is an angle bisector. 2. The circumcenter is equidistant to each of the triangles vertices. m∠ABD= m∠DBC

Jul 24­9:36 AM Jul 24­9:36 AM

The three angle bisectors of a triangle Incenter properties intersect at a single point. 1. The incenter is the The point of concurrency of the angle center of the bisectors is called the incenter. inscribed circle

Point A is the incenter of the triangle 2. The incenter is equidistant to each AB = AD = AC side of the triangle.

Jul 24­9:36 AM Jul 24­9:36 AM

An altitude is a segment from a The three altitudes of a triangle are vertex perpendicular to the opposite side concurrent. The point of concurrency is called the orthocenter.

m∠ADB= m∠ADC=90° Point A is the orthocenter of the triangle AD is an altitude of ∆ABC

Jul 24­9:36 AM Jul 24­9:36 AM 2 A median is a segment from a vertex The three medians of the triangle are to the midpoint of the opposite side concurrent. The point of concurrency is called the centroid.

AB is a median of ∆ACD Point A is the centroid of the triangle.

Jul 24­9:36 AM Jul 24­9:36 AM

3