`/_\ ABC` is an isosceles right triangle in which `/_A = 90^(@)` and BC = 6 cm. Then AB = ?

Options

A. `2sqrt2 cm`
B. `3sqrt2 cm`
C. `4sqrt2 cm`
D. `2sqrt3 cm`

Answer

To solve the problem, we need to find the length of side AB in the isosceles right triangle ABC, where angle A is 90 degrees and BC = 6 cm.

### Step-by-Step Solution:

1. **Identify the Triangle Properties**:
- Triangle ABC is an isosceles right triangle with angle A = 90°.
- In an isosceles right triangle, the two legs (sides opposite the equal angles) are equal. Therefore, AB = AC.

2. **Assign Variables**:
- Let AB = AC = x cm (since they are equal).
- The hypotenuse BC = 6 cm.

3. **Apply the Pythagorean Theorem**:
- According to the Pythagorean theorem, in a right triangle:
\[ \text{(Hypotenuse)}^2 = \text{(Base)}^2 + \text{(Perpendicular)}^2 \]
- Here, BC is the hypotenuse, and AB and AC are the legs:
\[ BC^2 = AB^2 + AC^2 \]
- Substituting the known values:
\[ 6^2 = x^2 + x^2 \]

4. **Simplify the Equation**:
- Calculate \(6^2\):
\[ 36 = x^2 + x^2 \]
- Combine like terms:
\[ 36 = 2x^2 \]

5. **Solve for x²**:
- Divide both sides by 2:
\[ x^2 = \frac{36}{2} = 18 \]

6. **Find x**:
- Take the square root of both sides:
\[ x = \sqrt{18} \]
- Simplify \(\sqrt{18}\):
\[ x = \sqrt{9 \cdot 2} = 3\sqrt{2} \]

7. **Conclusion**:
- Therefore, the length of AB is:
\[ AB = 3\sqrt{2} \text{ cm} \]