Chapters 3.7 – 3.8 “Nothing in life is to be feared, it is only to be understood.” Marie Cure

Objectives Discover points of concurrency of the angle bisectors, perpendicular bisectors, and altitudes of a triangle. Explore the relationships between points of concurrency and inscribed and circumscribed circles. Discover the concurrence of the medians of a triangle (the centroid) and its applications. Explore length relationships among the segments into which the centroid divides each median.

Vocabulary ____ Concurrent ____ Point of Concurrency ____ Incenter ____ Circumcenter ____ Orthocenter A) The point of concurrency for the three angle bisectors is the incenter. B) The point of concurrency for the three altitudes. C) The point of concurrency for the perpendicular bisector. D) The point of intersection E) Three or more line have a point in common. E D A C B

Vocabulary ____ Circumscribed ____ Inscribed ____ Centroid ____ Center of Gravity A) To draw (one figure) within another figure so that every vertex of the enclosed figure touches the outer figure. B) The balancing point for a polygon. C) To enclose a polygon within a configuration of lines, curves, or surfaces so that every vertex of the enclosed object is lying on the enclosing configuration. D) The point of concurrency of the three medians. C A D B

Conjectures Angle Bisector Concurrency Conjecture The three angle bisectors of a triangle ____________________. Perpendicular Bisector Concurrency Conjecture The three perpendicular bisectors of a triangle _____________. Altitude Concurrency Conjecture The three altitudes (or lines containing the altitudes) of a triangle _____________. meet at a point (are concurrent) are concurrent

Conjectures Circumcenter Conjecture The circumcenter of a triangle ____________________. Incenter Conjecture The incenter of a triangle __________________. Median Concurrency Conjecture The three medians of a triangle _____________. is equidistant from the vertices is equidistant from the sides are concurrent

Conjectures Centroid Conjecture The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is ______ the distance from the centroid to the midpoint of the opposite side. Center of Gravity Conjecture The _______ of a triangle is the center of gravity of the triangular region. twice centroid

Project Choose the project that you will do with your partner. The projects will be presented Wednesday and Thursday.