Answer:
Fourth Choice
Step-by-step explanation:
To answer this, we will start by using the Midpoint Theorem, which states that the segment that connects the midpoints of two sides of a triangle is parallel to the third side.
In this case, since D is midpont of AB and E is midpoint of BC, DE is parallel to AC.
Because they are parallel, the angles ∠BED and ∠BCA are corresponding angles and so are the angles ∠BDE and ∠BAC. Also, the angles ∠ABC and ∠DBC are the same, so:
∠BED=∠BCA
∠BDE=∠BAC
∠ABC= ∠DBC
Thus the triangles ΔABC and ΔBDE are similar.
This means that:
[tex]\frac{BC}{BE}=\frac{BA}{BD}=\frac{AC}{DE}=k [/tex]
We can calculate k using BE and BC:
[tex]\frac{BC}{BE}=k [/tex]
Since E is midpoint of BC, BC = 2*BE, so:
[tex]\frac{BC}{BE}=k [/tex]
[tex]\frac{2xBE}{BE}=k [/tex]
[tex]2=k\\
k=2[/tex]
Now, we can use the relation of DE, AC and k to calculate AC:
[tex]\frac{AC}{DE}=k=2\\
AC=2xDE\\
AC=2x3\\
AC=6[/tex]
This corresponds to the fourth alternative.
