Question

in $\triangle bcd$, $\overline{bc} \parallel \overline{ef}$. given that $db = 33$, $bc = 44$, and $ef = 20$, find $de$.

Explanation:

Step1: Identify Similar Triangles

Since \(\overline{BC} \parallel \overline{EF}\), by the Basic Proportionality Theorem (Thales' theorem), \(\triangle DEF \sim \triangle DBC\) (corresponding angles are equal, so triangles are similar).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{DE}{DB}=\frac{EF}{BC}\). Let \(DE = x\), then \(DB = DE + EB = x + EB\), but we can use the ratio directly as \(\frac{DE}{DB}=\frac{EF}{BC}\). Wait, actually, \(DB\) is the length of the side from \(D\) to \(B\), and \(DE\) is from \(D\) to \(E\), \(BC\) and \(EF\) are the bases. So the correct proportion is \(\frac{DE}{DB}=\frac{EF}{BC}\)? Wait, no. Wait, in similar triangles \(\triangle DEF\) and \(\triangle DBC\), the corresponding sides are \(DE\) corresponds to \(DB\), \(DF\) corresponds to \(DC\), and \(EF\) corresponds to \(BC\). Wait, no, actually, \(E\) is on \(DB\) and \(F\) is on \(DC\), so \(\triangle DEF \sim \triangle DBC\) with \(DE/DB = EF/BC\). Wait, let's check: \(DE\) is part of \(DB\), so \(DE = x\), \(DB = 33\), \(EF = 20\), \(BC = 44\). So the proportion is \(\frac{DE}{DB}=\frac{EF}{BC}\)? Wait, no, that would be if \(E\) is on \(DB\) and \(F\) is on \(DC\), so the sides \(DE\) and \(DB\) are corresponding, \(EF\) and \(BC\) are corresponding. So \(\frac{DE}{DB}=\frac{EF}{BC}\). Wait, but let's plug in the values: \(\frac{x}{33}=\frac{20}{44}\). Then solve for \(x\): \(x=\frac{20\times33}{44}\). Simplify: \(44\) divides into \(33\) as \(3/4\), so \(20\times\frac{3}{4}=15\). Wait, let's calculate: \(20\times33 = 660\), \(660\div44 = 15\). So \(DE = 15\).

Wait, let's verify the similar triangles again. Since \(EF \parallel BC\), \(\angle DEF = \angle DBC\) (corresponding angles) and \(\angle DFE = \angle DCB\) (corresponding angles), so \(\triangle DEF \sim \triangle DBC\) by AA similarity. Therefore, the ratio of corresponding sides is equal: \(\frac{DE}{DB}=\frac{EF}{BC}\). So substituting the values: \(\frac{DE}{33}=\frac{20}{44}\). Solving for \(DE\): \(DE=\frac{20\times33}{44}\). Simplify \(33/44 = 3/4\), so \(20\times3/4 = 15\). So \(DE = 15\).

Answer:

15