The Midline Theorem

• Midline theorem


  Segment with endpoints as midpoints of two sides of a triangle is parallel to the third side and half as long
• Proof of midline theorem

  1. Consider triangle BAC

  2. Point D is midpoint of BA

  3. Point N is midpoint of AC

  4. AE is congruent to EC

  5. Use vertical angle theorem to show triangle AED is congruent to triangle FEC

  6. Use SAS postulate to show BD is congruent to FC

  7. Use CPCTC to show BD is parallel to FC

  8. Therefore, BDFC is a parallelogram
• Midpoint


  Point that divides a line segment into two congruent parts
• If x and y are midpoints of BE and EA

  BX is congruent to EX, and BA is congruent to 2xy
• Finding length of LN in triangle LMN

  1. LN = x + 10, EF = x + 3.5

  2. 2EF = 2x + 7

  3. LN = 2EF = 13
• Finding length of FD in triangle LMN

  1. FD = 2SR, SR = 2x - 14

  2. FD = 2(2x - 14) = 4x - 28

  3. FD = 30
• Finding length of BD and AE in triangle ACE

  1. C = 19, D = 19, B = 21

  2. BD + AE = 2B = 42
• Finding length of BD and AE when BD = 2x - 1, AE = x + 4


  1. AE = 2BD

  2. x + 4 = 2(2x - 1)

  3. x = 2, BD = 3
• Finding length of BA when BA = 4a - 17, EB = 2a - 1


  1. EB = BA

  2. 2a - 1 = 4a - 17

  3. a = 8, BA = 15
• Finding length of DE when CA = 34, CE = 28

  1. CA + CE = 62

  2. CA = 2BA, CE = 2DE

  3. DE = 14