Question

If in two triangles ABC and DEF, \(\frac{AB}{EF} = \frac{BC}{DE} = \frac{CA}{DF}\); then :

• A. \(\Delta\)DEF \(\sim\) \(\Delta\)BCA
• B. \(\Delta\)DEF \(\sim\) \(\Delta\)CBA
• C. \(\Delta\)ABC \(\sim\) \(\Delta\)DEF
• D. \(\Delta\)ABC \(\sim\) \(\Delta\)DFE

Hint:

To find the correct vertex correspondence quickly:
Look at two ratios, say \(\frac{AB}{EF} = \frac{BC}{DE}\).
The common letter in the numerator is \(B\), and the common letter in the denominator is \(E\). Thus, vertex \(B\) corresponds to vertex \(E\).
Similarly, comparing \(\frac{BC}{DE} = \frac{CA}{DF}\), the common letter in the numerator is \(C\), and in the denominator is \(D\). Thus, \(C\) corresponds to \(D\).
This leaves the remaining vertex \(A\) to correspond to \(F\).
So, \(\Delta ABC \sim \Delta FED\) or \(\Delta DEF \sim \Delta CBA\).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem gives a ratio of corresponding sides of two triangles, \(\Delta ABC\) and \(\Delta DEF\), as \(\frac{AB}{EF} = \frac{BC}{DE} = \frac{CA}{DF}\). We need to determine the correct similarity notation that represents this relation.

Step 3: Detailed Explanation:
1. According to the SSS (Side-Side-Side) similarity criterion, if the corresponding sides of two triangles are in the same ratio, then their corresponding angles are equal, and the triangles are similar.
2. The notation for similarity must strictly reflect the one-to-one correspondence of the vertices.
3. Let us write down the given ratios of the sides:
\[ \frac{AB}{EF} = \frac{BC}{DE} = \frac{CA}{DF} \]
4. Let's analyze the vertices from the ratios:
- Look at the numerators (sides of \(\Delta ABC\)): \(AB\), \(BC\), and \(CA\).
- Look at the corresponding denominators (sides of \(\Delta DEF\)): \(EF\), \(DE\), and \(DF\).
5. Let's check Option (B): \(\Delta DEF \sim \Delta CBA\).
For this similarity to be true, the ratio of corresponding sides must be:
\[ \frac{DE}{CB} = \frac{EF}{BA} = \frac{DF}{CA} \]
Taking the reciprocal of each fraction, we get:
\[ \frac{CB}{DE} = \frac{BA}{EF} = \frac{CA}{DF} \]
Which is exactly:
\[ \frac{BC}{DE} = \frac{AB}{EF} = \frac{CA}{DF} \]
This is identical to the given relationship:
\[ \frac{AB}{EF} = \frac{BC}{DE} = \frac{CA}{DF} \]
Thus, the vertex correspondence is:
\[ D \leftrightarrow C \]
\[ E \leftrightarrow B \]
\[ F \leftrightarrow A \]
Therefore, \(\Delta DEF \sim \Delta CBA\) is the correct similarity relation.

Step 4: Final Answer:
The correct similarity statement is \(\Delta DEF \sim \Delta CBA\), which is option (B).