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Lesson 5-1 Bisectors, Medians, and Altitudes. Ohio Content Standards:

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Presentation on theme: "Lesson 5-1 Bisectors, Medians, and Altitudes. Ohio Content Standards:"— Presentation transcript:

1 Lesson 5-1 Bisectors, Medians, and Altitudes

2 Ohio Content Standards:

3 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 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 Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.

9 Example CD A B

10 Theorem 5.2

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

12 Example CD A B

13 Concurrent Lines

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

15 Point of Concurrency

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

17 Circumcenter

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

19 Theorem 5.3 Circumcenter Theorem

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

21 Example C A B circumcenter K

22 Theorem 5.4

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

24 Theorem 5.5

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

26 Incenter

27 The point of concurrency of the angle bisectors.

28 Theorem 5.6 Incenter Theorem

29 The incenter of a triangle is equidistant from each side of the triangle. C A Bincenter K P Q R

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

31 Median

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

33 Centroid

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

35 Theorem 5.7 Centroid Theorem

36 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 C A B DL E F centroid

38 Altitude

39 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 The intersection point of the altitudes of a triangle.

42 Example C A B D L E F orthocenter

43 Points U, V, and W are the midpoints of YZ, ZX, and XY, respectively. Find a, b, and c. Y W U X V Z 7.4 5c 8.7 15.2 2a2a 3b + 2

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


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