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Section 5-3 Concurrent Lines, Medians, and Altitudes

Section 5-3 Concurrent Lines, Medians, and Altitudes. B. D. F. C. A. E. 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.

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Section 5-3 Concurrent Lines, Medians, and Altitudes

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  1. Section 5-3 Concurrent Lines, Medians, and Altitudes

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

  3. B D F G C A E 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.

  4. B B E A C A C E 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.

  5. B E A C F Why?

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

  7. 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.

  8. 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.

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

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

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

  12. 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)

  13. 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)

  14. 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

  15. 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

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