In triangle `ABC, AB=6,AC=3sqrt6,angleB=60^@" and " angle C=45^(@)`. Find length of side BC.

Explanation

To find the length of side BC in triangle ABC where AB = 6 cm, AC = 3√6 cm, angle B = 60°, and angle C = 45°, we can follow these steps:

### Step 1: Identify the angles and sides
Given:
- AB = 6 cm
- AC = 3√6 cm
- Angle B = 60°
- Angle C = 45°

### Step 2: Calculate angle A
Using the property that the sum of angles in a triangle is 180°:
\[ \text{Angle A} = 180° - \text{Angle B} - \text{Angle C} = 180° - 60° - 45° = 75° \]

### Step 3: Draw a perpendicular from A to BC
Let AD be the perpendicular from A to side BC, where D is the foot of the perpendicular.

### Step 4: Calculate BD using triangle ABD
In triangle ABD, we can use the cosine of angle B:
\[ \cos(60°) = \frac{BD}{AB} \]
Substituting the known values:
\[ \frac{1}{2} = \frac{BD}{6} \]
Solving for BD:
\[ BD = 6 \cdot \frac{1}{2} = 3 \text{ cm} \]

### Step 5: Calculate CD using triangle ACD
In triangle ACD, we can use the cosine of angle C:
\[ \cos(45°) = \frac{CD}{AC} \]
Substituting the known values:
\[ \frac{1}{\sqrt{2}} = \frac{CD}{3\sqrt{6}} \]
Solving for CD:
\[ CD = 3\sqrt{6} \cdot \frac{1}{\sqrt{2}} = \frac{3\sqrt{6}}{\sqrt{2}} = 3 \cdot \sqrt{3} = 3\sqrt{3} \text{ cm} \]

### Step 6: Calculate BC
Now, we can find the length of side BC:
\[ BC = BD + CD = 3 + 3\sqrt{3} \]

### Final Answer
Thus, the length of side BC is:
\[ BC = 3(1 + \sqrt{3}) \text{ cm} \]

---