Medians, and Altitudes

When three or more lines intersect in one point, they are concurrent. The point at which they intersect is the point of concurrency. Point of concurrency For any triangle, there are four different sets of lines of concurrency.

Theorem 5-6 The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. A B C D

The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle. From the circumcenter, a circle can be circumscribed about the triangle. circumcenter

Theorem 5-7 The angle bisectors of a triangle are concurrent at a point equidistant from the sides. A B C D

The point of concurrency of the angle bisectors of a triangle is called the incenter of the triangle. incenter Using the incenter, a circle can be inscribed in a triangle.

A median of a triangle is a segment whose endpoints are a vertex and the midpoint of the opposite side. median

In a triangle, the point of concurrency of the medians is the centroid. The centroid is the center of gravity for a triangle – where it will balance. centroid

Theorem 5-8 The medians of a triangle are concurrent at a point that is two-thirds the distance from each vertex to the midpoint of the opposite side. F G D H E J C 2/3 1/3 2/3 1/3

F G D H E J C C is the circumcenter of ∆FDE. 1.Find DC if DJ = 15 2.Find CH if FH = 21 3.Find GE if GC = 6 4.Find DJ if DC = 18 DC = 10; CJ = 5 CH = 7; FC = 14 CE = 12; GE = 18 CJ = 9; DJ = 27

F G D H E J C C is the circumcenter of ∆FDE. Find x. DC = 4x - 12 CJ = x + 3

F G D H E J C C is the circumcenter of ∆FDE. Find x. FH = 15x + 6 CH = 3x + 12

An altitude of a triangle is the perpendicular segment from a vertex to the line containing the opposite side. An altitude can be inside the triangle, a side of a triangle, or it may lie outside the triangle. Acute - altitude is inside. Right - altitude is a leg. Obtuse - altitude is outside.

The altitudes are concurrent at the orthocenter of the triangle. Orthocenter

Summary 2/3 & 1/3 Vertex to midpoint Midpt of side & Perpendicular Vertex Vertex & Perpendicular

Is AB a perpendicular bisector, an altitude, a median, an angle bisector, or none of these? AltitudeMedian None

Is AB a perpendicular bisector, an altitude, a median, an angle bisector, or none of these? Perpendicular Bisector Angle Bisector Altitude