If `DeltaABC` is an isosceles triangle such that `angleABC=90^@`, then the true statement about `DeltaABC` is:

Options

A. `2(AC)^2=(AB)^2`
B. `(AB)^2+(AC)^2=(BC)^2`
C. `(BC)^2+(AC)^2=(AB)^2`
D. `(AC)^2=2(AB)^2`

Answer

To solve the problem, we need to analyze the properties of triangle ABC, which is given to be an isosceles triangle with angle ABC equal to 90 degrees.

### Step-by-Step Solution:

1. **Identify the Triangle Type**:
Triangle ABC is an isosceles triangle with angle ABC = 90 degrees. This means that the two sides opposite the equal angles (which are angles A and C) are equal.

2. **Label the Sides**:
Let AB = AC = x (since it's isosceles) and BC = y.

3. **Apply the Pythagorean Theorem**:
In a right triangle, the Pythagorean theorem states that the square of the hypotenuse (BC in this case) is equal to the sum of the squares of the other two sides (AB and AC):
\[ BC^2 = AB^2 + AC^2 \]
Since AB = AC, we can substitute:
\[ y^2 = x^2 + x^2 \]
This simplifies to:
\[ y^2 = 2x^2 \]

4. **Rearranging the Equation**:
From the equation \(y^2 = 2x^2\), we can express \(x^2\) in terms of \(y^2\):
\[ x^2 = \frac{y^2}{2} \]

5. **Identify the True Statement**:
The original question asks for a true statement about triangle ABC. From our derived equation, we can conclude:
\[ AC^2 = 2 \cdot AB^2 \]
Since we defined \(AB = AC = x\), we can write:
\[ AC^2 = 2 \cdot AB^2 \text{ (where AB = AC)} \]
This means the correct statement is:
\[ AC^2 = 2 \cdot AB^2 \]

### Conclusion:
The true statement about triangle ABC is:
\[ AC^2 = 2 \cdot AB^2 \]