Question:easy

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

Show 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 \(\Delta DEF \sim \Delta CBA\).
Updated On: Jul 7, 2026
  • \(\Delta\)DEF \(\sim\) \(\Delta\)BCA
  • \(\Delta\)DEF \(\sim\) \(\Delta\)CBA
  • \(\Delta\)ABC \(\sim\) \(\Delta\)DEF
  • \(\Delta\)ABC \(\sim\) \(\Delta\)DFE
Show Solution

The Correct Option is B

Solution and Explanation

Step 1: Understand what is being asked.
We are given $\frac{AB}{EF}=\frac{BC}{DE}=\frac{CA}{DF}$ for triangles $ABC$ and $DEF$, and we need the correct similarity statement. Instead of testing each option one by one, let us read off the vertex correspondence directly from the given ratios.

Step 2: Match vertices using the first two ratios.
Look at the equal ratios $\frac{AB}{EF}=\frac{BC}{DE}$. The letter common to both numerators (sides of triangle $ABC$) is $B$. The letter common to both denominators (sides of triangle $DEF$) is $E$.
Since in similar triangles matching sides come from matching vertices, this tells us:
\[ B \leftrightarrow E \]
Step 3: Match vertices using the next pair of ratios.
Now look at $\frac{BC}{DE}=\frac{CA}{DF}$. The letter common to both numerators is $C$, and the letter common to both denominators is $D$. This gives:
\[ C \leftrightarrow D \]
Step 4: Find the last correspondence and write the similarity.
The only vertex of triangle $ABC$ left unmatched is $A$, and the only vertex of triangle $DEF$ left unmatched is $F$, so:
\[ A \leftrightarrow F \]
Putting all three correspondences together, $A\to F$, $B\to E$, $C\to D$, we can write $\Delta ABC \sim \Delta FED$. Reading the vertices of $DEF$ in the order that matches $ABC$ reversed gives the equivalent, more natural statement:
\[ \Delta DEF \sim \Delta CBA \]
Final Answer:
The correct similarity is $\Delta DEF \sim \Delta CBA$, matching option (B).
\[ \boxed{\Delta DEF \sim \Delta CBA} \]
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