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Bisectors, Medians, and Altitudes: Exploring Geometric Figures

This lesson formally defines and explains key aspects of geometric figures, including interior angles, segments related to triangles, and points of concurrency. Topics covered include perpendicular bisectors, concurrent lines, circumcenter, incenter, medians, and altitudes. Theorems and examples are provided to illustrate these concepts.

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Bisectors, Medians, and Altitudes: Exploring Geometric Figures

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  1. Lesson 5-1 Bisectors, Medians, and Altitudes

  2. Ohio Content Standards:

  3. Ohio Content Standards: • Formally define geometric figures.

  4. Ohio Content Standards: • Formally define and explain key aspects of geometric figures, including:a. interior and exterior angles of polygons;b. segments related to triangles (median, altitude, midsegment);c. points of concurrency related to triangles (centroid, incenter, orthocenter, and circumcenter);

  5. Perpendicular Bisector

  6. Perpendicular Bisector A line, segment, or ray that passes through the midpoint of the side of a triangle and is perpendicular to that side.

  7. Theorem 5.1

  8. Theorem 5.1 Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.

  9. Example A C D B

  10. Theorem 5.2

  11. Theorem 5.2 Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.

  12. Example A C D B

  13. Concurrent Lines

  14. Concurrent Lines When three or more lines intersect at a common point.

  15. Point of Concurrency

  16. Point of Concurrency The point of intersection where three or more lines meet.

  17. Circumcenter

  18. Circumcenter The point of concurrency of the perpendicular bisectors of a triangle.

  19. Theorem 5.3Circumcenter Theorem

  20. Theorem 5.3Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of the triangle.

  21. Example B circumcenter K A C

  22. Theorem 5.4

  23. Theorem 5.4 Any point on the angle bisector is equidistant from the sides of the angle. B C A

  24. Theorem 5.5

  25. Theorem 5.5 Any point equidistant from the sides of an angle lies on the angle bisector. B C A

  26. Incenter

  27. Incenter The point of concurrency of the angle bisectors.

  28. Theorem 5.6Incenter Theorem

  29. Theorem 5.6Incenter Theorem The incenter of a triangle is equidistant from each side of the triangle. incenter B Q K P A C R

  30. Theorem 5.6Incenter Theorem The incenter of a triangle is equidistant from each side of the triangle. incenter B If K is the incenter of ABC, then KP = KQ = KR. Q K P A C R

  31. Median

  32. Median A segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex.

  33. Centroid

  34. Centroid The point of concurrency for the medians of a triangle.

  35. Theorem 5.7Centroid Theorem

  36. Theorem 5.7Centroid Theorem The centroid of a triangle is located two-thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median.

  37. Example B E D L centroid A C F

  38. Altitude

  39. Altitude A segment from a vertex in a triangle to the line containing the opposite side and perpendicular to the line containing that side.

  40. Orthocenter

  41. Orthocenter The intersection point of the altitudes of a triangle.

  42. Example B E D orthocenter L A C F

  43. Points U, V, and W are the midpoints of YZ, ZX, and XY, respectively. Find a, b, and c. Y 7.4 W U 5c 8.7 3b + 2 15.2 2a X Z V

  44. The vertices of QRS are Q(4, 6), R(7, 2), and S(1, 2). Find the coordinates of the orthocenter of QRS.

  45. Assignment:Pgs. 243-245 13-20 all, 23-30 all

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