Concurrent Lines Medians and Altitiudes Concurrent Lines When

Concurrent Lines, Medians, and Altitiudes

Concurrent Lines • When three or more lines intersect in one point, they are concurrent. • Point of Concurrency – the point at which the lines intersect. • For a triangle, there are four different sets of concurrent lines.

Circumcenter • The perpendicular bisectors of the sides of a triangle are concurrent (all intersect) at a point equidistant from the vertices. – The point of concurrency is called the circumcenter of the triangle.

Picture of the Circumcenter H CH = CJ = CG C is the circumcenter. C G Using a compass with the point at C, a circle can be drawn that passes thru G, H and J. The circle is circumscribed about the triangle. J

Incenter • The bisectors of the angles of a triangle are concurrent (all intersect) at a point equidistant from the sides. – The point of concurrency is called the incenter of the triangle.

Picture of the Incenter IY = IR = IW T I is the incenter. Y I Q W R E Using a compass with the point at I, a circle can be drawn that passes thru Y, R and W. The circle is inscribed in the triangle.

Medians • Median of a Triangle – A median is a segment drawn from the vertex of a triangle to the midpoint of the opposite side. Median

Centroid • The medians of a triangle are concurrent (all intersect) at a point that divides each median into two segments, one of which is twice as long as the other. – The point of concurrency is called the centroid. – Also called the center of gravity or center of balance.

Picture of Centroid 2 x 8 k 5 2 b C x 4 b 2 k 10

Altitudes • Altitude of a Triangle – An altitude is a segment drawn from the vertex of a triangle perpendicular to the opposite side. – Can be inside, on a side, or outside the traingle Acute Triangle: Right Triangle: Obtuse Triangle: Altitude inside Altitude is a side Altitude outside

Orthocenter • The lines that contain the altitudes of a triangle are concurrent (they all intersect at a single point). – The point of concurrency of the altitudes is called the orthocenter.

Orthocenter

Finding Lengths of Medians • M is the centroid of DWOR, and WM = 16. Find WX. WX = 24

Identifying Medians and Altitudes • Is KX a median, an altitude, neither, or both? Both