ae = ec and bf = fc. ef = 8 and df = 14.
ab = ?

Question

Accepted Answer

# Explanation:
## Step1: Identify Midsegments
Since $ AE = EC $ and $ BF = FC $, $ E $ is midpoint of $ AC $, $ F $ is midpoint of $ BC $. So $ EF $ is midsegment of $ \triangle ABC $, $ DF $ is midsegment? Wait, no, $ D $ and $ E $: Wait, $ AE = EC $, $ BF = FC $, and $ DE $ and $ EF $? Wait, actually, in triangle, midsegment theorem: segment connecting midpoints of two sides is parallel to third side and half its length. Wait, $ E $ is midpoint of $ AC $, $ F $ is midpoint of $ BC $, so $ EF \parallel AB $ and $ EF=\frac{1}{2}AB $? Wait, no, maybe $ D $ is midpoint? Wait, $ DF $ and $ EF $: Wait, $ AE = EC $, so $ E $ is midpoint of $ AC $. $ BF = FC $, so $ F $ is midpoint of $ BC $. Then $ EF $ is midsegment of $ \triangle ABC $, so $ EF \parallel AB $ and $ EF = \frac{1}{2}AB $? Wait, no, maybe $ D $ is midpoint of $ AB $? Wait, $ DF $ and $ EF $: Wait, $ DE $ and $ DF $. Wait, the problem: $ EF = 8 $, $ DF = 14 $. Wait, maybe $ DE $ is midsegment? Wait, no, let's re-examine.

Wait, $ AE = EC $ (so $ E $ is midpoint of $ AC $) and $ BF = FC $ (so $ F $ is midpoint of $ BC $). Then $ EF $ is midsegment of $ \triangle ABC $, so $ EF \parallel AB $ and $ EF = \frac{1}{2}AB $? Wait, no, midsegment connects midpoints of two sides, so midpoint of $ AC $ and midpoint of $ BC $ would connect to midpoint of $ AB $? Wait, no, midsegment between $ AC $ and $ BC $ would be parallel to $ AB $ and half its length. So $ EF \parallel AB $ and $ EF = \frac{1}{2}AB $? But then $ AB = 2 \times EF = 16 $? But wait, $ DF = 14 $. Wait, maybe $ D $ is midpoint of $ AB $, so $ DF $ is midsegment? Wait, $ F $ is midpoint of $ BC $, $ D $ is midpoint of $ AB $, so $ DF \parallel AC $ and $ DF = \frac{1}{2}AC $. But $ E $ is midpoint of $ AC $, so $ DE \parallel BC $ and $ DE = \frac{1}{2}BC $. Wait, maybe the figure is a parallelogram? Wait, $ AE = EC $, $ BF = FC $, and $ D $, $ E $, $ F $: maybe $ DEFB $ is parallelogram? Wait, no, let's think again.

Wait, the problem is to find $ AB $. Given $ EF = 8 $, $ DF = 14 $. Wait, maybe $ EF $ is midsegment, so $ AB = 2 \times EF $? But that would be 16, but what about $ DF $? Wait, maybe $ D $ is midpoint, so $ AD = DB $, and $ E $ is midpoint, so $ DE \parallel BC $, and $ F $ is midpoint, so $ DF \parallel AC $, making $ DEFC $ a parallelogram? Wait, no, let's use midsegment theorem properly.

Midsegment theorem: In a triangle, the segment connecting the midpoints of two sides is parallel to the third side and half as long.

So, $ E $ is midpoint of $ AC $ (since $ AE = EC $), $ F $ is midpoint of $ BC $ (since $ BF = FC $). Therefore, $ EF $ is the midsegment of $ \triangle ABC $, so $ EF \parallel AB $ and $ EF = \frac{1}{2}AB $. Wait, but then $ AB = 2 \times EF = 2 \times 8 = 16 $. But what about $ DF = 14 $? Maybe that's a distractor, or maybe I misidentified. Wait, no, maybe $ D $ is midpoint of $ AB $, so $ AD = DB $, and $ F $ is midpoint of $ BC $, so $ DF $ is midsegment of $ \triangle ABC $, so $ DF \parallel AC $ and $ DF = \frac{1}{2}AC $. But $ E $ is midpoint of $ AC $, so $ AC = 2 \times DF = 28 $, but that's not needed. Wait, the question is $ AB $, so using $ EF $ as midsegment: $ EF \parallel AB $, $ EF = \frac{1}{2}AB $, so $ AB = 2 \times 8 = 16 $.

Wait, maybe the figure has $ D $ as midpoint of $ AB $, so $ AD = DB $, $ E $ as midpoint of $ AC $, so $ AE = EC $, so $ DE \parallel BC $, and $ F $ as midpoint of $ BC $, so $ DF \parallel AC $, making $ DEFC $ a parallelogram, but the key is $ EF $ is midsegment for $ AB $. So:

## Step1: Apply Midsegment Theorem
Since $ E $ (midpoint of $ AC $) and $ F $ (midpoint of $ BC $) are midpoints, $ EF $ is midsegment of $ \triangle ABC $. By midsegment theorem, $ EF = \frac{1}{2}AB $.

## Step2: Solve for $ AB $
Given $ EF = 8 $, so $ AB = 2 \times EF = 2 \times 8 = 16 $.

Wait, but what about $ DF = 14 $? Maybe that's for another part, but the question is $ AB $, so using $ EF $ as midsegment.

# Answer:
$ \boxed{16} $

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QUESTION IMAGE

Question

Explanation:

Step1: Identify Midsegments

Answer: