For finding the area of a triangle ABC, which of the following entities are required ?

Options

A. Angles A, B and side a
B. Angles A, B and side b
C. Angles A, B and side c
D. Either (a) or (b) or (c)

Explanation

To find the area of triangle ABC, we can use various formulas depending on the information we have. Here’s a step-by-step explanation of the required entities:

### Step 1: Understanding the Area of a Triangle
The area of a triangle can be calculated using different formulas based on the given parameters. The most common formulas include:
1. **Using base and height**: Area = (1/2) × base × height
2. **Using two sides and the included angle**: Area = (1/2) × a × b × sin(C), where a and b are the lengths of two sides, and C is the included angle.
3. **Using Heron’s formula**: Area = √[s(s-a)(s-b)(s-c)], where s is the semi-perimeter of the triangle, and a, b, c are the lengths of the sides.

### Step 2: Identifying Required Entities
To determine which entities are required to find the area of triangle ABC, we need to consider the options provided:

1. **Option 1: Angle A, Angle B, and Side A**
- Here, we have two angles and one side. However, we cannot directly find the area because we need either the height or another side to use the angle-side-angle formula.

2. **Option 2: Angle A, Angle B, and Side B**
- Similar to option 1, this option also provides two angles and one side. We still lack the necessary information to calculate the area.

3. **Option 3: Angle A, Angle B, and Side C**
- This option provides two angles and one side. However, we can use the sine rule or the formula for the area involving two sides and the included angle. Since we have two angles, we can find the third angle (Angle C) and then use the side C with one of the angles to calculate the area.

### Conclusion
Based on the analysis, the correct entities needed to find the area of triangle ABC are:
- **Angle A, Angle B, and Side C** (Option 3).