Question

tutorial exercise
in the figure below, \\( \overline{ac} \parallel \overline{de} \\), \\( bd \\) measures 7 m, \\( ad \\) measures 14 m, and \\( be \\) measures 8 m. find the length of \\( \overline{bc} \\).

triangle figure with vertices a, b, c and d on ab, e on bc, de parallel to ac

part 1 of 4
first, find the length of the segment \\( \overline{ab} \\). substitute the values for \\( ad \\) and \\( db \\) and simplify.
\\( ab = ad + bd \\)
\\( ab = 14 + 7 \\)
\\( = 21 \\)

part 2 of 4
triangles \\( abc \\) and \\( dbe \\) are similar. therefore the ratios of the corresponding sides are equal. complete the equation relating the corresponding sides.
\\( \frac{ab}{bd} = \frac{bc}{\square} \\)

Explanation:

Step1: Recall Similar Triangles Property

Since $\triangle ABC \sim \triangle DBE$ (by Basic Proportionality Theorem or AA similarity as $AC \parallel DE$), the ratios of corresponding sides are equal. Corresponding sides: $AB$ corresponds to $DB$, and $BC$ corresponds to $BE$.

Step2: Identify Corresponding Sides Ratio

For similar triangles $\triangle ABC$ and $\triangle DBE$, the ratio $\frac{AB}{BD}$ should equal the ratio of the other corresponding sides $\frac{BC}{BE}$. So the missing term in the denominator of the right - hand side ratio (with numerator $BC$) should be $BE$.

Answer:

$\frac{BE}{1}$ (or just $BE$, but in the boxed form as per the equation structure, the answer is $\frac{BC}{BE}$ so the denominator is $BE$. So the value to put in the box is $BE$? Wait, no, the equation is $\frac{AB}{BD}=\frac{BC}{x}$, and since corresponding sides, $AB$ (from $\triangle ABC$) corresponds to $DB$ (from $\triangle DBE$), and $BC$ (from $\triangle ABC$) corresponds to $BE$ (from $\triangle DBE$). So $\frac{AB}{BD}=\frac{BC}{BE}$. So the blank should be $BE$. But if we consider the structure, the right - hand side is $\frac{BC}{BE}$, so the denominator is $BE$. So the answer for the blank is $BE$? Wait, no, the equation is $\frac{AB}{BD}=\frac{BC}{?}$, and since $\triangle ABC \sim \triangle DBE$, $\frac{AB}{DB}=\frac{BC}{BE}$. So the missing part is $BE$. So the answer is $BE$. But let's check again. The triangles are similar, so $\frac{AB}{DB}=\frac{BC}{BE}$. So the equation $\frac{AB}{BD}=\frac{BC}{BE}$, so the blank is $BE$. So the answer is $BE$.