Triangle Centers and Lines Quiz

Triangle Centers and Lines

• Centroid: Intersection of medians; located 2/3 of the distance from each vertex to the midpoint of the opposite side. Used to find the center of mass.
• Incenter: Intersection of angle bisectors; equidistant from all three sides. Used to find the center of the inscribed circle.
• Circumcenter: Intersection of perpendicular bisectors; equidistant from all three vertices. Used to find the center of a circumscribed circle.
• Orthocenter: Intersection of altitudes; has unique properties related to the triangle's angles.

Types of Lines and Centers

• Median: Connects a vertex to the midpoint of the opposite side. Center: Centroid
• Altitude: Perpendicular segment from a vertex to the opposite side. Center: Orthocenter
• Perpendicular Bisector: Perpendicular to a side and through its midpoint. Center: Circumcenter
• Angle Bisector: Divides an angle into two equal angles. Center: Incenter

Theorems

• Perpendicular Bisector Theorem: A point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
• Converse of Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment..
• Angle Bisector Theorem: A point on the angle bisector is equidistant from the sides of the angle.
• Converse of Angle Bisector Theorem: If a point is equidistant from the sides of an angle, then it is on the angle bisector.

Triangle Properties

• Concurrent Lines: Lines that intersect at a single point.
• Midsegment: Connects midpoints of two sides of a triangle.
• Acute Triangle: Altitude inside.
• Right Triangle: Altitude is a side.
• Obtuse Triangle: Altitude outside.
• Exterior Angle Theorem: Measure of an exterior angle is greater than either remote interior angle.

Point of Concurrency

• A point of concurrency is a point where three (or more) lines intersect.

Circumcenter specific

• The circumcenter is located on the hypotenuse of a right triangle.