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Properties of Medians and Altitudes in Triangles

This lesson covers the properties of medians and altitudes in triangles, explaining the concept of a median as a segment from a vertex to the midpoint of the opposite side. Theorem 5-8 highlights that medians are concurrent at the centroid, the triangle's center of gravity. Examples demonstrate how to find lengths of medians and identify medians and altitudes. Additionally, the definition of altitudes is provided, emphasizing their potential location inside or outside the triangle. Theorem 5-9 discusses the concurrency of altitude lines at the orthocenter.

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Properties of Medians and Altitudes in Triangles

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  1. Chapter 5 Lesson 3 Objective: To identify properties of medians and altitudes of a triangle.

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

  3. Theorem 5-8 The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side.

  4. In a triangle, the point of concurrency of the medians is the centroid. • The point is also called the center of gravity of a triangle because it is the point where a triangular shape will balance.

  5. Example 1: Finding Lengths of Medians D is the centroid of ∆ABC and DE = 6. Find BE. Since D is a centroid, BD =  BE and DE = BE.

  6. Example 2: Finding Lengths of Medians Use Example 1 to find BD. BD =  BE

  7. An altitude of a triangle is the perpendicular segment from a vertex to the line containing the opposite side. Unlike angle bisectors and medians, an altitude of a triangle can be a side of a triangle or it may lie outside the triangle.

  8. Example 3: Identifying Medians and Altitudes Is ST a median, an altitude, or neither? Explain. ST is an altitude; perpendicular segment from a vertex

  9. Example 4: Identifying Medians and Altitudes Is UW a median, an altitude, or neither? Explain. Median; UW is a segment drawn from vertex U to the midpoint of the opposite side.

  10. The lines containing the altitudes of a triangle are concurrent at the orthocenter of a triangle. Theorem 5-9 The lines that contain the altitudes of a triangle are concurrent.

  11. Assignment:

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