Mid-Segment Theorem in Triangles

Introduction

• Discussion on the mid-segment theorem for triangles.
• Mid-segment theorem relates midpoints of sides of a triangle.

Understanding Mid-Segment

1. Finding Midpoints

   • Point D: Midpoint of line segment AB.
   • Point E: Midpoint of line segment BC.
   • Point F: Midpoint of line segment AC.

2. Line Joining Midpoints

   • A line joining midpoints of two sides (e.g. DE, DF, FE) of a triangle.
   • This line is known as the mid-segment.

Mid-Segment Theorem

• Definition: A line segment joining midpoints of two sides of a triangle is:
  • Parallel to the third side.
  • Half the length of the third side.

Example Application

• Line DE

  • Connects midpoints of AB and BC.
  • DE is parallel to AC and DE = 1/2 AC.
  • DE is congruent to AF and CF.

• Line DF

  • Connects midpoints of AB and AC.
  • DF is parallel to BC and DF = 1/2 BC.
  • DF is congruent to BE and CE.

• Line FE

  • Connects midpoints of AC and BC.
  • FE is parallel to AB and FE = 1/2 AB.
  • FE is congruent to AD and DB.

Solving Problems

• Given DE = 30:

  • DE is parallel to BC and DE = 1/2 BC.
  • Compute BC: BC = 2 * DE = 60.

• Algebraic Example

  • Given DE as 2x + 5 and BC as 5x + 2.
  • Use relationships and solve simultaneous equations to find x.*

Conclusion

• Key idea: Mid-segment connects midpoints of two sides, parallel and half the length of the third side.
• Useful in solving geometric problems concerning triangles.