Question

In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

Hint:

Always double-check the segment additions along a straight line transversal.
Calculating $EF = 20\text{ cm}$ correctly is key, as using $BC$ or $EB$ incorrectly in the ratio would lead to erroneous scale factors!

Answer 1

Step 1: Prove similarity using corresponding angles.
Since $AB \parallel DE$ with transversal $EBCF$, corresponding angles give $\angle ABC = \angle DEF$. Since $AC \parallel DF$ with the same transversal, $\angle ACB = \angle DFE$. By AA similarity, $\Delta ABC \sim \Delta DEF$.
Step 2: Find the full length EF. \[ EF = EB + BC + CF = 5+10+5 = 20\text{ cm} \]
Step 3: Use a scale factor instead of solving a proportion equation.
The scale factor from $\Delta ABC$ to $\Delta DEF$ is $k = \frac{EF}{BC} = \frac{20}{10} = 2$. Since $DE$ corresponds to $AB$, it scales by the same factor $k$.
Step 4: Compute DE. \[ DE = AB \times k = 7 \times 2 = 14\text{ cm} \]
\[ \boxed{DE = 14\text{ cm}} \]