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
- We are given that ADADAD is the median of △ABC△ABC△ABC and EEE is the midpoint of AD.AD.AD.
Let us draw a line DGDGDG parallel to BFBFBF. - Now, in △ADG△ADG△ADG, EEE is the midpoint of ADADAD and EF∥DG.EF∥DG.EF∥DG.
By converse of the midpoint theorem we have FFF as midpoint of AG.AG.AG. [Math Processing Error]
Similarly, in △BCF△BCF, DD is the midpoint of BCBC and DG∥BF.DG∥BF.
By converse of midpoint theorem we have GG is midpoint of CF.CF. [Math Processing Error] - From equations (1) and (2), we get [Math Processing Error] Also, from the figure we see that [Math Processing Error]
- We are given that AFAF = 3 cm.
Thus, AC=3AF=3×3 cm=9 cmAC=3AF=3×3 cm=9 cm.