Question

It is given that \(\Delta ABC \sim \Delta EDF\). Which of the following is not true ?

• A. \(\frac{\text{Perimeter of } \Delta ABC}{\text{Perimeter of } \Delta EDF} = \frac{AB}{ED}\)
• B. \(\frac{AB}{ED} = \frac{AC}{EF}\)
• C. \(\angle A = \angle D, \angle C = \angle F\)
• D. \(\frac{AB + BC}{AC} = \frac{DE + DF}{EF}\)

Hint:

To avoid mistakes in similar triangle questions, write down the vertex correspondence explicitly before reading the options:
Map \(A \leftrightarrow E\), \(B \leftrightarrow D\), and \(C \leftrightarrow F\).
This simple step prevents visual confusion and helps you spot the incorrect angle mapping instantly!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The topic of this question is Similar Triangles.
When two triangles are similar, their corresponding angles are equal, and their corresponding sides are proportional.
The vertex correspondence in the similarity statement \(\Delta ABC \sim \Delta EDF\) is crucial because it defines the exact mapping between vertices, angles, and sides of the two triangles.
We need to analyze each given option to determine which statement is not mathematically true based on this correspondence.

Step 2: Key Formula or Approach:
From the similarity relation \(\Delta ABC \sim \Delta EDF\), we can establish the following:
1. Corresponding Angles are equal:
\[ \angle A = \angle E \] \[ \angle B = \angle D \] \[ \angle C = \angle F \] 2. Corresponding Sides are proportional:
\[ \frac{AB}{ED} = \frac{BC}{DF} = \frac{AC}{EF} \] 3. The ratio of the perimeters of two similar triangles equals the ratio of their corresponding sides:
\[ \frac{\text{Perimeter of } \Delta ABC}{\text{Perimeter of } \Delta EDF} = \frac{AB}{ED} \] We will evaluate each option against these criteria.

Step 3: Detailed Explanation:

• Check Option (A):
Since the ratio of perimeters equals the ratio of corresponding sides, and \(AB\) corresponds to \(ED\), the statement \(\frac{\text{Perimeter of } \Delta ABC}{\text{Perimeter of } \Delta EDF} = \frac{AB}{ED}\) is mathematically true.

• Check Option (B):
From the side proportionality, we have \(\frac{AB}{ED} = \frac{AC}{EF}\) because \(AB\) maps to \(ED\) and \(AC\) maps to \(EF\). Thus, this statement is true.

• Check Option (C):
The statement asserts that \(\angle A = \angle D\) and \(\angle C = \angle F\).
However, the vertex correspondence shows that vertex \(A\) maps to vertex \(E\), which means \(\angle A = \angle E\).
Vertex \(B\) maps to vertex \(D\), which means \(\angle B = \angle D\).
Therefore, the assertion \(\angle A = \angle D\) is not true.

• Check Option (D):
Let us assume the ratio of corresponding sides is equal to a constant \(\lambda\):
\[ \frac{AB}{ED} = \frac{BC}{DF} = \frac{AC}{EF} = \lambda \] This gives \(AB = \lambda \cdot ED\), \(BC = \lambda \cdot DF\), and \(AC = \lambda \cdot EF\).
Substitute these expressions into the left-hand side:
\[ \frac{AB + BC}{AC} = \frac{\lambda \cdot ED + \lambda \cdot DF}{\lambda \cdot EF} = \frac{\lambda(ED + DF)}{\lambda \cdot EF} = \frac{ED + DF}{EF} \] Since the left-hand side simplifies to match the right-hand side, this statement is true.


Step 4: Final Answer:
The statement that is not true is Option (C).
Therefore, the correct option is (C).