In an isosceles triangle `ABC, angleB= 90^@` and side `AB= 6 cm`. Side AC will be

Options

A. 6 cm
B. 12 cm
C. `2sqrt6` cm
D. `6sqrt2` cm

Answer

To find the length of side AC in the isosceles triangle ABC where angle B is 90 degrees and side AB is 6 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Triangle Properties**:
- Triangle ABC is isosceles with angle B = 90 degrees.
- This means that sides AB and BC are equal since the triangle is isosceles.

2. **Assign Known Values**:
- Given that AB = 6 cm, we can conclude that BC = 6 cm as well (because AB = BC in an isosceles triangle).

3. **Use the Pythagorean Theorem**:
- In a right triangle, the Pythagorean theorem states that:
\[ AC^2 = AB^2 + BC^2 \]
- Substitute the known values:
\[ AC^2 = 6^2 + 6^2 \]

4. **Calculate the Squares**:
- Calculate \(6^2\):
\[ 6^2 = 36 \]
- Therefore:
\[ AC^2 = 36 + 36 = 72 \]

5. **Find AC**:
- To find AC, take the square root of both sides:
\[ AC = \sqrt{72} \]
- Simplify \(\sqrt{72}\):
\[ \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} \]

6. **Final Answer**:
- Thus, the length of side AC is:
\[ AC = 6\sqrt{2} \text{ cm} \]