In a triangle ABCABCABC, ADADAD is a median. FFF is a point on ACACAC such that the line BFBFBF bisects ADADAD at EEE. If AD=9 cmAD=9 cmAD=9 cm and AF=3 cmAF=3 cmAF=3 cm, find the measure of ACACAC.


Answer:

9 cm

Step by Step Explanation:
  1. We are given that ADADAD is the median of ABCABCABC and EEE is the midpoint of AD.AD.AD.

    Let us draw a line DGDGDG parallel to BFBFBF.
      B C D G F A E


  2. Now, in ADGADGADG, EEE is the midpoint of ADADAD and EFDG.EFDG.EFDG.

    By converse of the midpoint theorem we have FFF as midpoint of AG.AG.AG. [Math Processing Error]

    Similarly, in BCFBCF, DD is the midpoint of BCBC and DGBF.DGBF.

    By converse of midpoint theorem we have GG is midpoint of CF.CF. [Math Processing Error]
  3. From equations (1) and (2), we get [Math Processing Error] Also, from the figure we see that [Math Processing Error]
  4. We are given that AFAF = 3 cm.
    Thus, AC=3AF=3×3 cm=9 cmAC=3AF=3×3 cm=9 cm.

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