Objectives: To define points of concurrency in triangles 5.3 (day 2) Concurrency Objectives: To define points of concurrency in triangles

Definitions Concurrent: when three or more lines intersect in one point, they are said to be concurrent Point of concurrency: the point at which the lines intersect For any triangle, there are four different sets of concurrent lines.

Points of Concurrency Incenter: the point of concurrency of the angle bisectors of a triangle Circumcenter: the point of concurrency of the perpendicular bisectors of a triangle Orthocenter: the point of concurrency of the altitudes of the triangle Centroid: the point of concurrency of the medians of a triangle

Points of Concurrency

Concurrency Theorems The circumcenter is equidistant from the vertices of the triangle. The incenter is equidistant from the sides of the triangle.

Definitions Inscribed: A circle is inscribed in a polygon if the sides of the polygon are tangent to the circle

Circumscribed: A circle is circumscribed about a polygon if the vertices of the polygon are on the circle.

Incenter of a Triangle The incenter of a triangle is the center of the circle that is inscribed in it.

Circumcenter of a Triangle The circumcenter of a triangle is the center of a circle that is circumscribed about it.

Example Find the center of the circle circumscribed around the triangle with vertices (0, 0), (-8, 0), (0, 6).

Theorem The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint.

Example If WM = 16, then WX = ____ If RZ = 30, then MZ = ____ If MY = 3, then MO = ____ W Z Y M R O X

Center of Gravity The centroid of a triangle is the center of mass or also the center of gravity. A triangle could be balanced on its centroid.

Assignment Page 259 #1-4, 8-9, 11-16, 19-22, 27-29, 37-39