Question

ae = ec and bf = fc. ba = 18, de = 7, and bc = 14.
fe = ?

Explanation:

Step1: Identify Midsegments

Since \( AE = EC \) and \( BF = FC \), \( E \) is the midpoint of \( AC \) and \( F \) is the midpoint of \( BC \). By the Midline Theorem (also known as the Midsegment Theorem) in triangles, the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. Here, \( FE \) connects midpoints \( F \) (of \( BC \)) and \( E \) (of \( AC \)), so \( FE \parallel AB \) and \( FE=\frac{1}{2}AB \).

Step2: Calculate \( FE \)

Given \( BA = 18 \), using the Midline Theorem formula \( FE=\frac{1}{2}BA \), we substitute \( BA = 18 \) into the formula: \( FE=\frac{1}{2}\times18 = 9 \).

Answer:

9