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Chapter 5

Chapter 5. More Triangles. Mr. Thompson. More Triangles. Mr. Thompson. Midsegment Theorem. A midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle. Midsegment Theorem.

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Chapter 5

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  1. Chapter 5 More Triangles. Mr. Thompson More Triangles. Mr. Thompson.

  2. Midsegment Theorem

  3. A midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle.

  4. Midsegment Theorem The segment connecting the midpoints of 2 sides of a triangle is parallel to the 3rd side and is ½ as long.

  5. Perpendiculars and Bisectors

  6. In 1.5, you learned that a segment bisector intersects a segment at its midpoint. midpoint 10 10 Segment bisector

  7. A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. Perpendicular bisector d f 12 12

  8. Y is equidistant from X and Z. A point is equidistant from two points if its distance from each point is the same. x z y

  9. Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. A 8 8 B

  10. Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, x z y Y is equidistant from X and Z.

  11. Using Perpendicular Bisectors What segment lengths in the diagram are equal? T 12 NS=NT (given) M is on the perpendicular bisector of ST, so….. MS=MT (Theorem 5.1) QS =QT=12 (given) N Q M 12 S

  12. Using Perpendicular Bisectors Explain why Q is on MN. QS=QT, so Q is equidistant from S and T. T 12 Q N M By Theorem 5.2, Q is on the perpendicular bisector of ST, which is MN. 12 S

  13. The distance from a point to a line….. defined as the length of the perpendicular segment from the point to the line. R The distance from point R to line m is the length of RS. m S

  14. Point that is equidistant from two lines… When a point is the same distance from one line as it is from another line, the point is equidistant from the two lines(or rays or segments).

  15. Angle Bisector Theorem • If a point is on the bisector of an angle, then it is equidistant from the 2 sides of the angle. • If angle ABD = angle CBD, then DC = AD.

  16. Converse of the Angle Bisector Theorem • If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.

  17. Classwork… Page 246: 13, 31, 32, 36, 38 (use graph paper) Page 252: 28, 29, 32, 33, 40, 46

  18. Bisectors of a Triangle…

  19. When three or more lines (or rays or segments) intersect in the same point, they are called concurrent lines (or rays or segments). The point of intersection of the lines is called the point of concurrency.

  20. Special Triangle Segments Both perpendicular bisectors and angle bisectors are often associated with triangles, as shown below. Triangles have two other special segments. Perpendicular Bisector Angle Bisector

  21. The three perpendicular bisectors of a triangle are concurrent. The point of concurrency can be inside the triangle, on the triangle, or outside the triangle.

  22. Circumcenter The point of concurrency of the three perpendicular bisectors of a triangle is called the circumcenterof the triangle. In each diagram, the circle circumscribesthe triangle.

  23. Incenter Concurrency of Angle Bisectors of a Triangle Theorem The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.

  24. Incenter The point of concurrency of the three angle bisectors of a triangle is called the incenterof the triangle. In the diagram, the circle is inscribedwithin the triangle.

  25. Points of Concurrency Summary Circumcenter: Perpendicular Bisectors Incenter: Angle Bisectors

  26. Medians and Altitudes

  27. A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.

  28. Median

  29. Median A medianof a triangle is a segment from a vertex to the midpoint of the opposite side of the triangle.

  30. Concurrency of Medians Theorem The medians of a triangle intersect at a point that is two-thirds of the distance from each vertex to the midpoint of the opposite side.

  31. Centroid The three medians of a triangle are concurrent. The point of concurrency is an interior point called the centroid. It is the balancing point or center of gravity of the triangle.

  32. An altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side. An altitude can lie inside, on or outside the triangle.

  33. Altitude

  34. Altitude An altitudeof a triangle is a perpendicular segment from a vertex to the opposite side or to the line that contains that side. The length of the altitude is the height of the triangle.

  35. Orthocenter Concurrency of Altitudes of a Triangle Theorem The lines containing the altitudes of a triangle are concurrent. G

  36. Orthocenter The point of concurrency of all three altitudes of a triangle is called the orthocenterof the triangle. The orthocenter, P, can be inside, on, or outside of a triangle depending on whether it is acute, right, or obtuse, respectively.

  37. 5.3 Classwork Page 260: 19-22, 27-29 35 (use whole sheet of paper)

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