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5.4 Medians and Altitudes

5.4 Medians and Altitudes. Objectives. Use properties of medians Locate the centroid Use properties of altitudes Locate the orthocenter. Vocabulary. A. A median is a segment from a vertex of a ∆ to the midpoint of the opposite side. Every ∆ has three medians.

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5.4 Medians and Altitudes

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  1. 5.4 Medians and Altitudes

  2. Objectives • Use properties of medians • Locate the centroid • Use properties of altitudes • Locate the orthocenter

  3. Vocabulary A • A medianis a segment from a vertex of a ∆ to the midpoint of the opposite side. Every ∆ has three medians. • These medians intersect at a common point, the point of concurrency, called the centroid. • Thecentroidis the point of balance for a ∆. MEDIAN C D B

  4. Theorem 5.8 • Theorem 5.8 (Centroid Theorem)The centroid of a ∆ is located two thirds of the distance from a vertex to the midpoint of the opposite side.

  5. In RST, Qis the centroid and SQ = 8. Find QWand SW. SQ = SW 2 2 3 3 8= SW 3 2 Multiply each side by the reciprocal, . 12= SW 12 – 8 = 4. SW – SQ = Then QW = Example 1 Concurrency of Medians of a Triangle Theorem Substitute 8 for SQ. So, QW = 4 and SW = 12.

  6. In Exercises 1–3, use the diagram. Gis the centroid of ABC. ANSWER ANSWER ANSWER 13.5 12 18 More Practice 1. If BG = 9, find BF. 2. If BD = 12, find AD. 3. If CD = 27, find GC.

  7. Locating the Centroid with Coordinates • When given 3 coordinates of a triangle and asked to locate the centroid, find the averages of the x-values, and the average of the y-values. This new coordinate will be the center of balance of the triangle, or the coordinate of the centroid. • Example: • Given triangle with following coordinates: • (3, 0) • (5, -2) • (4, 8) • Centroid x-value = (3 + 5 + 4)/3 • Centroid y-value = (0 + -2 +8)/3 • Centroid = (12/3, 6/3) or (4, 2)

  8. Vocabulary • An altitude of a ∆ is a segment from a vertex to the line containing the opposite side and is ┴ to that side. Every ∆ has three altitudes. • The intersection point, point of concurrency, of the altitudes of a ∆ is called the orthocenter.

  9. Diagrams of Previous Vocabulary Altitudes Orthocenter

  10. Additional Notes Orthocenter • Acute Triangle: Orthocenter is inside the triangle. • Right Triangle: Orthocenter is on the triangle. • Obtuse Triangle: Orthocenter is outside the triangle.

  11. Assignment • Workbooks Pg. 94 – 96 #1 – 25

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