3. If \(\triangle ABC \sim \triangle EDF\) and \(\triangle ABC\) is not similar to \(\triangle DEF\), which is not true?
\(BC\cdot EF = AC\cdot FD\)
\(AB\cdot EF = AC\cdot DE\)
\(BC\cdot DE = AB\cdot EF\)
\(BC\cdot DE = AB\cdot FD\)
Step 1: We are told that \(\triangle ABC \sim \triangle EDF\). This means the triangles are similar, so their corresponding sides are in the same ratio.
Step 2: Write the rule of similarity: \[ \dfrac{AB}{ED} = \dfrac{BC}{DF} = \dfrac{AC}{EF} \]
Step 3: Now, let’s test each option one by one.
(A) \(BC \cdot EF = AC \cdot FD\) From the ratio \(\dfrac{BC}{DF} = \dfrac{AC}{EF}\), cross-multiplying gives \(BC \cdot EF = AC \cdot DF\). This is true.
(B) \(AB \cdot EF = AC \cdot DE\) From the ratio \(\dfrac{AB}{ED} = \dfrac{AC}{EF}\), cross-multiplying gives \(AB \cdot EF = AC \cdot ED\). This is true.
(C) \(BC \cdot DE = AB \cdot EF\) To check this, look at the ratios: \(\dfrac{AB}{ED}\) and \(\dfrac{BC}{DF}\). These do not directly give \(BC \cdot DE = AB \cdot EF\). So ❌ this is not always true.
(D) \(BC \cdot DE = AB \cdot FD\) From the ratio \(\dfrac{AB}{ED} = \dfrac{BC}{DF}\), cross-multiplying gives \(BC \cdot ED = AB \cdot DF\). This is true.
Final Answer: Option (C) is not true.