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Understanding Medians in Triangles: Properties and Examples

This chapter explores the concept of medians in triangles, defining a median as a segment connecting a triangle's vertex to the midpoint of the opposite side. It explains the properties of medians, including their role in splitting sides into equal parts and their intersection point, called the centroid. Key examples illustrate how to find lengths related to medians, such as the relationship between segments created by the centroid and the vertices of the triangle. Additionally, practice problems reinforce these concepts.

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Understanding Medians in Triangles: Properties and Examples

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  1. Chapter 6: More about Triangles6-1: Medians

  2. 6-1: Medians • Median: A segment that joins a vertex of the triangle at the midpoint of the side opposite that vertex. • The median splits the side into two equal parts

  3. 6-1: Medians F • Example 1 • In EFG, FN is a median. • Find EN if EG = 11. • If FN is a median, then N isa midpoint of EG • If EG = 11, then EN is (left) half • EN = ½  11 = 5.5 N E G

  4. 6-1: Medians • Your Turn • In MNP, MC and ND are medians. • What is NC if NP = 18? • 9 • If DP = 7.5, find MP. • 15 • If PD = 7x – 1, CP = 5x – 4, and DM = 6x + 9, find NC. • 46

  5. 6-1: Medians • Centroid: The point where all three medians of a triangle intersect. • Concurrent: When three or more lines or segments meet at the same point. • X is the centroidof JKM. • QM, JR, and PKare concurrent.

  6. 6-1: Medians • There is a unique relationship between the length of a segment from the centroid to the vertex and from the centroid to the midpoint. • See the examples below.

  7. 6-1: Medians • Theorem 6-1: The length of a segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint. • Note this means: • Centroid to midpoint = 1/3 of whole median • Centroid to vertex = 2/3 of whole median

  8. 6-1: Medians • Example 2 • In XYZ, XP, ZN, and YM are medians. • Find ZQ if QN = 5. • ZQ is centroid to vertex • It’s twice as long as centroid to midpoint • ZQ = 2  5 = 10 • If XP = 10.5, what is QP? • QP is centroid to midpoint • It’s half the length of centroid to vertex (no good) • It’s 1/3rd the entire length of the median • QP = 1/3 10.5 = 3.5

  9. 6-1: Medians • Your Turn • In ABC, AE, BF and CD are medians. • If CG = 14, what is DG? • 7 • Find BF if GF = 6.8? • 20.4

  10. 6-1: Medians • Assignment • Worksheet #6-1

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