If in `DeltaABC,angleB=90^(@),AB=6sqrt3` and `AC=12cm`, find BC.

Options

A. 5 cm
B. 6 cm
C. 7 cm
D. 8 cm

Answer

To solve the problem, we will use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Given:
- Triangle ABC with angle B = 90°
- Side AB = 6√3 cm
- Side AC (hypotenuse) = 12 cm
- We need to find side BC.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
- In triangle ABC, angle B is the right angle.
- AB is one of the legs (6√3 cm).
- AC is the hypotenuse (12 cm).
- BC is the other leg, which we need to find.

2. **Apply the Pythagorean theorem**:
According to the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the known values:
\[ 12^2 = (6\sqrt{3})^2 + BC^2 \]

3. **Calculate the squares**:
- Calculate \(12^2\):
\[ 12^2 = 144 \]
- Calculate \((6\sqrt{3})^2\):
\[ (6\sqrt{3})^2 = 6^2 \cdot (\sqrt{3})^2 = 36 \cdot 3 = 108 \]

4. **Set up the equation**:
Now substituting these values back into the equation:
\[ 144 = 108 + BC^2 \]

5. **Isolate \(BC^2\)**:
To find \(BC^2\), subtract 108 from both sides:
\[ BC^2 = 144 - 108 \]
\[ BC^2 = 36 \]

6. **Find \(BC\)**:
Take the square root of both sides:
\[ BC = \sqrt{36} = 6 \text{ cm} \]

### Final Answer:
The length of side BC is **6 cm**.