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: Understanding the Question:
We are given a geometric figure where points $E, B, C, F$ lie on a straight line.
The segments $AB \parallel DE$ and $AC \parallel DF$.
We need to show that $\Delta ABC$ is similar to $\Delta DEF$ and then find the length of side $DE$.

Step 2: Key Formula or Approach:
We will use the Angle-Angle (AA) similarity criterion to prove similarity.
Once similarity is established, the ratio of corresponding sides of similar triangles must be equal:
\[ \frac{AB}{DE} = \frac{BC}{EF} \]

Step 3: Detailed Explanation:

• Establish similarity between $\Delta ABC$ and $\Delta DEF$:
- Consider the line segment $EF$ as a transversal cutting parallel lines $AB$ and $DE$.
Since $AB \parallel DE$, corresponding angles are equal:
\[ \angle ABC = \angle DEF \]
- Consider the line segment $EF$ as a transversal cutting parallel lines $AC$ and $DF$.
Since $AC \parallel DF$, corresponding angles are equal:
\[ \angle ACB = \angle DFE \]
- By the Angle-Angle (AA) similarity criterion:
\[ \Delta ABC \sim \Delta DEF \]

• Find the total length of side $EF$:
We are given that $E, B, C, F$ are collinear in that sequence.
\[ EF = EB + BC + CF \]
Substitute the given values ($BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$):
\[ EF = 5 + 10 + 5 = 20\text{ cm} \]

• Apply the property of corresponding sides in similar triangles:
\[ \frac{AB}{DE} = \frac{BC}{EF} \]
Substitute the known values ($AB = 7\text{ cm}$, $BC = 10\text{ cm}$, $EF = 20\text{ cm}$):
\[ \frac{7}{DE} = \frac{10}{20} \]
\[ \frac{7}{DE} = \frac{1}{2} \]

• Solve for $DE$:
\[ DE = 7 \times 2 = 14\text{ cm} \]


Step 4: Final Answer:
The triangles are similar ($\Delta ABC \sim \Delta DEF$), and the length of $DE$ is $14\text{ cm}$.