Section 5 3 Concurrent Lines Medians and Altitudes

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Section 5 -3 Concurrent Lines, Medians, and Altitudes

Section 5 -3 Concurrent Lines, Medians, and Altitudes

Triangle Medians A median of a triangle is a line segment drawn from any

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

Triangle Medians Theorem B F A D G E The medians of a triangle

Triangle Medians Theorem B F A D G E The medians of a triangle are concurrent at a point (called the centroid) that is two thirds the distance from each C vertex to the midpoint of the opposite side.

Triangle Altitudes An altitude of a triangle is a line segment drawn from any

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. B B A E C E A C

B Why? E A F C

B Why? E A F C

Triangle Altitude Theorem The lines that contain the altitudes of a triangle are concurrent

Triangle Altitude Theorem The lines that contain the altitudes of a triangle are concurrent (at a point called the orthocenter). B A E C

Triangle Perpendicular Bisectors Theorem The perpendicular bisectors of a triangle are concurrent at a

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 X Q C Z Y R The circle is circumscribed about the triangle.

Triangle Angle Bisectors Theorem The bisectors of the angles of a triangle are concurrent

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 X U I Z Y V The circle is inscribed in the triangle.

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

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

Application U X In triangle TUV, Y is the centroid. YW=9. Find TY and

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

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

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

Application Find the center of the circle you can circumscribe about the triangle with

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

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 S • Draw a

Try these constructions: 1: Circumscribe a circle about a triangle S • 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. X Q C Z Y R

Try these constructions: 2: Construct a circle inside a triangle T • Draw a

Try these constructions: 2: Construct a circle inside a triangle T • 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 U 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. X I Z Y V