In `Delta ABC angle B = 60^(@) angle C = 45^(@) AB = 12` cm Find area of `Delta `

Answer

To find the area of triangle ABC given that angle B = 60°, angle C = 45°, and side AB = 12 cm, we can follow these steps:

### Step 1: Find Angle A
We know that the sum of the angles in a triangle is 180°. Therefore, we can find angle A using the formula:
\[ \text{Angle A} = 180° - \text{Angle B} - \text{Angle C} \]
Substituting the given values:
\[ \text{Angle A} = 180° - 60° - 45° = 75° \]

### Step 2: Use the Area Formula for a Triangle
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times a \times b \times \sin(C) \]
Where \( a \) and \( b \) are the lengths of two sides, and \( C \) is the included angle between those sides. In our case, we can use:
- \( a = AB = 12 \) cm
- To find \( AC \) and \( BC \), we will use the sine rule.

### Step 3: Use the Sine Rule to Find Other Sides
The sine rule states:
\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]
Let’s denote:
- \( a = BC \)
- \( b = AC \)
- \( c = AB = 12 \) cm

Using the sine rule:
\[ \frac{BC}{\sin(75°)} = \frac{12}{\sin(45°)} \]
From this, we can find \( BC \):
\[ BC = \frac{12 \times \sin(75°)}{\sin(45°)} \]

### Step 4: Calculate the Lengths of Sides
Now, we can calculate \( BC \):
Using \( \sin(75°) \approx 0.9659 \) and \( \sin(45°) \approx 0.7071 \):
\[ BC = \frac{12 \times 0.9659}{0.7071} \approx 16.36 \text{ cm} \]

### Step 5: Calculate Area Using Two Sides and Included Angle
Now we can calculate the area using \( AB \) and \( BC \) with angle A:
\[ A = \frac{1}{2} \times AB \times BC \times \sin(A) \]
Substituting the values:
\[ A = \frac{1}{2} \times 12 \times 16.36 \times \sin(75°) \]
Using \( \sin(75°) \approx 0.9659 \):
\[ A \approx \frac{1}{2} \times 12 \times 16.36 \times 0.9659 \]
Calculating this gives:
\[ A \approx \frac{1}{2} \times 12 \times 16.36 \times 0.9659 \approx 94.25 \text{ cm}^2 \]

### Final Answer
The area of triangle ABC is approximately \( 94.25 \text{ cm}^2 \).

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