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Transcript
3.1 Special Segments and Centers of Triangles I CAN... Define and recognize perpendicular bisectors, angle bisectors, medians, and altitudes. Define and recognize points of concurrency. Jul 249:36 AM Classifications of Triangles: By angle 1. Acute: A triangle with three acute angles. Classifications of Triangles: By Side: 1. Equilateral: A triangle with three congruent sides. 2. Isosceles: A triangle with at least two congruent sides. 3. Scalene: A triangle with three sides having different lengths. (no sides are congruent) Jul 249:36 AM Special Segments and Centers in Triangles A Perpendicular Bisector is a segment or line that passes through the midpoint of a side and 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. When three or more lines intersect at the same point, it is called a "Point of Concurrency." Jul 249:36 AM The point of concurrency of the perpendicular bisectors is called the circumcenter. Jul 249:36 AM 1 Circumcenter Properties 1. The circumcenter is the center of the circumscribed circle. An angle bisector is a segment that divides an angle into two congruent angles. BD is an angle bisector. 2. The circumcenter is equidistant to each of the triangles vertices. Jul 249:36 AM The three angle bisectors of a triangle intersect at a single point. The point of concurrency of the angle bisectors is called the incenter. Point A is the incenter of the triangle Jul 249:36 AM An altitude is a segment from a vertex perpendicular to the opposite side m∠ABD= m∠DBC Jul 249:36 AM Incenter properties 1. The incenter is the center of the inscribed circle 2. The incenter is equidistant to each side of the triangle. AB = AD = AC Jul 249:36 AM The three altitudes of a triangle are 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 to the midpoint of the opposite side The three medians of the triangle are 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