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Geometry

Learn to identify and use medians and altitudes in triangles, including finding centroids. Homework included.

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Geometry

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  1. Geometry Lesson 5 – 2 Medians and Altitudes of Triangles Objective: Identify and use medians in triangles. Identify and use attitudes in triangles.

  2. Median • Median of a triangle • A segment with endpoints at a vertex of a triangle and the midpoint of the opposite side.

  3. Centroid • Centroid • The point of concurrency of the medians of a triangle. • Centroid Theorem • The medians of a triangle intersect at a point called the centroid that is two thirds of the distance from each vertex to the midpoint of the opposite side.

  4. Centroid… PK = 5 Find AP 10 x 2x BP = 12 Find PL. 6 JC = 15 Find JP. PK + AP = AK 5 PK + 2(PK) = AK

  5. In triangle ABC, Q is the centroid and BE = 9 • Find BQ • Find QE BQ = 2(QE) = 3 OR 6 = 2(QE) 3 = QE

  6. In triangle ABC, Q is the centroid and FC = 14 • Find FQ • Find QC = 5 QC = 2(FQ) QC = 10 QC = 2(5)

  7. In triangle JKL, PT = 2. Find KP. How do you know that P is the centroid? KP = 2(PT) = 2(2) = 4 OR KP = 4

  8. In triangle JKL, RP = 3.5 and JP = 9 PL = 2(RP) • Find PL • Find PS = 2(3.5) = 7 PS = 4.5 JP = 2(PS) 9 = 2(PS)

  9. A performance artist plans to balance triangular pieces of metal during her next act. When one such triangle is placed on the coordinate plane, its vertices are located at (1, 10) (5, 0) and (9,5). What are the coordinates of the point where the artist should support the triangle so that it will balance. The balance point of a triangle is the centroid.

  10. Hint: To make it easier look for a vertical or horizontal line between a midpoint of a side and vertex. • Graph the points. Find the midpoint of the side(s) that could make a vertical or horizontal line. Find the midpoint of AB. Midpoint of AB = = (3, 5) Let P be the Centroid, where would it be? From the vertex to the centroid is 2/3 of the whole.

  11. Count over from C 4 units and that is P Centroid: (5, 5)

  12. A second triangle has vertices (0,4), (6, 11.5), and (12,1). What are the coordinates of the point where the artist should support the triangle so that it will balance? Explain your reasoning. Centroid: (6, 5.5)

  13. Altitude • Altitude of a triangle • A perpendicular segment from a vertex to the side opposite that vertex. Draw a right triangle and identify all the altitudes.

  14. Orthocenter • Orthocenter • The lines containing the altitudes of a triangle are concurrent, intersecting at a point called the orthocenter.

  15. Find the orthocenter • The vertices of triangle FGH are F(-2, 4), G(4,4), and H(1, -2). Find the coordinates of the orthocenter of triangle FGH. Graph the points. Cont…

  16. m = 2 Slope of GH. New equation is perpendicular to segment GH. Point F (-2, 4) m = -1/2 • Find an equation from F to GH. y = mx + b 3 = b Cont…

  17. m = -2 Slope of segment FH New equation is perpendicular to segment FH. Point G (4, 4) m = 1/2 • Find an equation from G to FH. 2 = b Cont…

  18. If the orthocenter lies on an exact point of the graph use the graph to name. If it does not lie on a point use systems of equations to find the orthocenter. • The orthocenter can be found at the intersection of our 2 new equations. • How can we find the orthocenter? System of equations: Cont…

  19. By substitution. Orthocenter (1, 2.5) 1 = x y = 2.5

  20. Summary • Perpendicular bisector

  21. Summary • Angle bisector

  22. Summary • Median

  23. Summary • Altitude

  24. Homework • Pg. 337 1 – 10 all, 12 - 20 E, 27 – 30 all, 48 – 54 E

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