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. A B C D E F Question: If I printed this slide in black and white, what would be incorrect about the figure?

Triangle Medians Theorem 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. A B C D E F G

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. Q S R C X Y 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. U T V I X Y Z The circle is inscribed in the triangle.

Application O W R Y X Z M M is the centroid of triangle WOR. WM=16. Find WX. WX=24

Application T U V W Z X Y In triangle TUV, Y is the centroid. YW=9. Find TY and TW. TW=27 TY=18

Application L K M X Is KX a median, altitude, neither, or both? both

Application Find the center of the circle you can circumscribe about the triangle with vertices: A (-4, 5); B (-2, 5); C (-2, -2) (-3, 1.5) Hint: sketch triangle; then think about the perpendicular bisectors passing through the midpoints of the sides!

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. Q S R C X Y 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. U T V I X Y Z