Section 5-3 Concurrent Lines, Medians, and Altitudes
Triangle Medians A median of a triangle is a line segment drawn from any vertex of the triangle to the midpoint of the opposite side. B D F C A E
Centroid Theorem A B C D E F G The medians of a triangle are concurrent at a point (called the centroid) that is two thirds the distance from each vertex to the midpoint of the opposite side.
Triangle Altitudes An altitude of a triangle is a line segment drawn from any vertex of the triangle to the opposite side, extended if necessary, and perpendicular to that side. A B C E A B C E
A B C E F Why?
Triangle Altitude Theorem The lines that contain the altitudes of a triangle are concurrent (at a point called the orthocenter). A B C E
Triangle Perpendicular Bisectors Theorem The perpendicular bisectors of a triangle are concurrent at a point (called the circumcenter) that is equidistant from the vertices. S Y X C Q R Z The circle is circumscribed about the triangle.
Triangle Angle Bisectors Theorem The bisectors of the angles of a triangle are concurrent at a point (called the incenter) that is equidistant from the sides. T Y X I U V Z The circle is inscribed in the triangle.
M is the centroid of triangle WOR. WM=16. Find WX. Application W M is the centroid of triangle WOR. WM=16. Find WX. Y WX=24 Z M R O X
In triangle TUV, Y is the centroid. YW=9. Find TY and TW. Application In triangle TUV, Y is the centroid. YW=9. Find TY and TW. U TY=18 W TW=27 X Y V T Z
Is KX a median, altitude, neither, or both? Application K both L M X
Application Find the center of the circle you can circumscribe about the triangle with vertices: A (-4, 5); B (-2, 5); C (-2, -2) Hint: sketch triangle; then think about the perpendicular bisectors passing through the midpoints of the sides! (-3, 1.5)
Application Find the center of the circle you can circumscribe about the triangle with vertices: X (1, 1); Y (1, 7); Z (5, 1) (3, 4)
Try these constructions: 1: Circumscribe a circle about a triangle Draw a large triangle. Construct the perpendicular bisectors of any two sides. The point they meet is the circumcenter. The radius is from the circumcenter to one of the vertices. Draw a circle using this radius and it should pass through all three vertices. S Y X C Q R Z
Try these constructions: 2: Construct a circle inside a triangle Draw a large triangle. Construct the angle bisectors for two of the angles. The point they intersect is called the incenter. Drop a perpendicular from the incenter to one of the sides. This is your radius. Draw a circle using this radius and it should touch each side of the triangle. T Y X I U V Z