A statements of Assertion (A) is followed by a statement of Reason (R ). Mark the correct choice as.
Assertion (A): ABC is an isoseles triangle right triangle , right angled at C. Then `AB^(2) = 3AC^(2)`.
Reason (R ): In an isosceles triangle ABC if AC = BC and `AB^(2) = 2AC^(2)` then `angle C = 90^(@)`.

Options

A. Both A and R are true R is the correct explanation for A.
B. Both A and R are true nad R is not correct explanation for A.
C. A is true but R is false.
D. A is false but R is true.

Answer

To solve the problem, we need to analyze both the Assertion (A) and the Reason (R) provided in the question.

### Step-by-Step Solution:

1. **Understanding the Assertion (A)**:
- The assertion states that triangle ABC is an isosceles triangle right-angled at C, and claims that \( AB^2 = 3AC^2 \).

2. **Understanding the Reason (R)**:
- The reason states that in an isosceles triangle ABC, if \( AC = BC \) and \( AB^2 = 2AC^2 \), then \( \angle C = 90^\circ \).

3. **Analyzing the Assertion**:
- Since triangle ABC is right-angled at C and isosceles, we have \( AC = BC \).
- By the Pythagorean theorem, we know:
\[ AB^2 = AC^2 + BC^2 \]
- Since \( AC = BC \), we can substitute:
\[ AB^2 = AC^2 + AC^2 = 2AC^2 \]
- Thus, the assertion \( AB^2 = 3AC^2 \) is incorrect because we derived \( AB^2 = 2AC^2 \).

4. **Analyzing the Reason**:
- The reason states that if \( AC = BC \) and \( AB^2 = 2AC^2 \), then \( \angle C = 90^\circ \).
- This is indeed true as it follows directly from the Pythagorean theorem, confirming that if the sum of the squares of the two equal sides equals the square of the hypotenuse, then the angle opposite the hypotenuse must be \( 90^\circ \).

5. **Conclusion**:
- The assertion (A) is **false** because \( AB^2 \) does not equal \( 3AC^2 \) but rather \( 2AC^2 \).
- The reason (R) is **true** as it correctly describes a property of isosceles right triangles.

### Final Answer:
- Assertion (A) is false.
- Reason (R) is true.