Special Segments of Triangles

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# Special Segments of Triangles - PowerPoint PPT Presentation

Special Segments of Triangles. Sections 5.2, 5.3, 5.4. Perpendicular bisector theorem. A point is on the perpendicular bisector if and only if it is equidistant from the endpoints of the segment. Angle Bisector Theorem.

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### Special Segments of Triangles

Sections 5.2, 5.3, 5.4

Perpendicular bisector theorem
• A point is on the perpendicular bisector if and only if it is equidistant from the
• endpoints of the segment.
Angle Bisector Theorem
• A point is on the bisector of an angle if and only if it is equidistant from the two sides of the angle.
Medians of a triangle
• A median of a triangle is a segment from a vertex to the midpoint of the
• opposite side.
Altitudes of a triangle
• An altitude of a triangle is the perpendicular segment from a vertex to the
• opposite side or to the line that contains the opposite side.
Concurrency
• The point of intersection of the lines, rays, or segments is called the point of concurrency.
Points of concurrency
• The point of concurrency of the three perpendicular bisectors a triangle is called the circumcenter.
• The point of concurrency of the three angle bisectors of a triangle is called the incenter.
• The point of concurrency of the three medians of a triangle is called the centroid.
• The point of concurrency of the three altitudes of a triangle is called the orthocenter.
• The incenter and centroid will always be inside the triangle. The circumcenter
• and orthocenter can be inside, on, or outside the triangle.
What is special about the Circumcenter?
• The perpendicular bisectors of a triangle intersect at a point that is equisdistant
• from the vertices of the triangle.

PA = PB = PC

What is special about the Incenter?
• The angle bisectors of a triangle intersect at a point that is equidistant from the
• sides of the triangle.

PD = PE = PF

What is special about the Centroid?
• The medians of a triangle intersect at a point that is two thirds of the distance
• from each vertex to the midpoint of the opposite side.
What is special about the Orthocenter?
• There is nothing special about the point of concurrency of the altitudes of a
• triangle.