5-3 Points of Concurrency Objective: To identify properties of perpendicular bisectors and angle bisectors

Concurrent: When three or more lines, segments, rays or planes have a point in common. Point of Concurrency: The point of intersection. Not Concurrent Concurrent Point of Concurrency

The three perpendicular bisectors of a triangle are concurrent. Point of concurrency is called circumcenter. The circumcenter of a triangle is equidistant from the vertices and is the center of the circle.

The circumcenter of a triangle can be inside, on, or outside a triangle.

Finding the Circumcenter of a triangle.

The point of concurrency of the angle bisectors of a triangle is called the incenter of the triangle. The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle. P is the center of the circle that is inscribed in the triangle. p. 305: 7, 8, 9, 15, 17, 37, 38

The circle is circumscribed about the triangle. Circumcenter

The three angle bisectors of a triangle are concurrent. Point of concurrency is called incenter. Angle Bisector Concurrency Conjecture Incenter Conjecture The incenter of a triangle is equidistant from the sides. Inscribed Circle Incenter

Altitude Concurrency Conjecture The three altitudes (or the lines containing the altitudes) of a triangle are concurrent. Point of concurrency is called orthocenter. Pg 179 # 1-4, 6 Orthocenter

GroupConstruct 1Each angle bisectors for an acute triangle 2Each angle bisectors for an obtuse triangle 3Each perpendicular bisectors for an acute triangle 4Each perpendicular bisectors for an obtuse triangle 5Each altitude for an acute triangle 6Each altitude for an obtuse triangle Directions 1.Each group member does assigned construction. 2.Compare and discuss all constructions 3.Write a conjecture about your construction 4.Measure from point of concurrency to vertices and sides 5.Discuss findings and write a conjecture. 6.Elect spokes person to share construction and conjectures