In triangle ABC, segment DE is parallel to side BC, with D on AB and E on AC. If AD=4, DB=6, and EC=9, what is AE?

Because $DE\parallel BC$, the side-splitter theorem says the line divides the two sides proportionally: $\frac{AD}{DB}=\frac{AE}{EC}$. Substituting, $\frac{4}{6}=\frac{AE}{9}$, so $AE=\frac{4\cdot9}{6}=6$, choice B. Check using the full sides: $AB=10$ and $AC=15$, and $\frac{4}{10}=\frac{6}{15}$. Choice D ($13.5$) inverts the proportion, $\frac{6\cdot9}{4}$. Choices A ($5$) and C ($7.5$) come from setting up the proportion with the wrong pairs of segments. The proportion must compare the top parts to the bottom parts on both sides.