Points B, D, and F are midpoints of the sides of △ACE. EC = 38 and DF = 16. Find AC. The diagram is not to scale.
We know that B, D, and F are midpoints of the sides of triangle ACE. This means that the segments BD, DF, and FB are all parallel to their respective sides of the triangle.
Since DF = 16, and FB is parallel to AC, we can conclude that AB = 2(DF) = 2(16) = 32.
Similarly, since BD is parallel to AE, we know that BD = AE = (1/2)AC.
Since B is the midpoint of AE, we know that AB = BE.
Therefore, AB + BE = AE = 32.
But AB = 32, so we can substitute that in to get 32 + BE = 32.
Subtracting 32 from both sides gives us BE = 0.
Since E is the midpoint of AC, we know that AE = EC.
Therefore, AE + EC = AC = 2(EC) = 2(38) = 76.
So, AC = <<76=76>>76.