To determine the length of EC, the Basic Proportionality Theorem (Thales' theorem) is applied. This theorem states that a line parallel to one side of a triangle proportionally divides the other two sides.
Since DE is parallel to BC, the theorem yields:
\(\frac{AD}{DB} = \frac{AE}{EC}\)
The given values are:
\(AD = 6\, \text{cm}, \quad DB = 4\, \text{cm}, \quad AE = 9\, \text{cm}\)
Substituting these values into the equation results in:
\(\frac{6}{4} = \frac{9}{EC}\)
Cross-multiplication gives:
\(6 \times EC = 9 \times 4\)
\(6 \times EC = 36\)
To find EC, both sides are divided by 6:
\(EC = \frac{36}{6}\)
\(EC = 6\, \text{cm}\)