Question

Which of the following statements is not always true?

• A. Two circles are similar.
• B. Two isosceles right triangles are similar.
• C. Two rectangles are similar.
• D. Two equilateral triangles are similar.

Hint:

For regular polygons of the same number of sides (like equilateral triangles or squares), similarity is guaranteed.
Rectangles are not regular polygons because their adjacent sides do not have to be equal, meaning their aspect ratios can differ.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to identify which of the geometric statements regarding similarity is not always true.
Two geometric figures are similar if they have the exact same shape, meaning their corresponding angles are equal and their corresponding sides are in the same ratio.

Step 2: Key Formula or Approach:
We will analyze each option based on geometric similarity criteria: - Circles: Always have a constant circular shape.
- Isosceles Right Triangles: Angles are always \(45^\circ\), \(45^\circ\), and \(90^\circ\).
- Equilateral Triangles: Angles are always \(60^\circ\), \(60^\circ\), and \(60^\circ\).
- Rectangles: All angles are \(90^\circ\), but the ratio of adjacent sides (length to width) can vary.

Step 3: Detailed Explanation:
1. Option (A): Two circles are similar.
Any two circles have the same shape. The ratio of their radii is the scale factor of similarity. Thus, all circles are always similar. This statement is true.
2. Option (B): Two isosceles right triangles are similar.
Every isosceles right triangle has angles of \(90^\circ\), \(45^\circ\), and \(45^\circ\). Since the corresponding angles of any two isosceles right triangles are always equal, they are always similar by AAA similarity. This statement is true.
3. Option (C): Two rectangles are similar.
For two rectangles to be similar, their corresponding sides must be proportional. While all rectangles have equal interior angles (\(90^\circ\)), their side lengths can have different ratios. For example, a rectangle with sides \(2\text{ cm} \times 4\text{ cm}\) is not similar to a rectangle with sides \(3\text{ cm} \times 9\text{ cm}\) because: \[ \frac{2}{3} \neq \frac{4}{9} \] Thus, two rectangles are not always similar. This statement is not always true.
4. Option (D): Two equilateral triangles are similar.
Every equilateral triangle has interior angles of \(60^\circ\), \(60^\circ\), and \(60^\circ\). By AAA similarity, any two equilateral triangles are always similar. This statement is true.

Step 4: Final Answer:
The statement "Two rectangles are similar" is not always true because their corresponding sides are not necessarily proportional.
Hence, the correct option is (C).