The area bound by the relation `|x|+|y|=2` is

Options

A. 8
B. 1
C. 2
D. 4

Explanation

To find the area bounded by the relation \( |x| + |y| = 2 \), we can follow these steps:

### Step 1: Understand the Equation
The equation \( |x| + |y| = 2 \) represents a diamond (or rhombus) shape in the coordinate plane. This is because the absolute values create linear segments in each quadrant.

### Step 2: Identify Intercepts
To find the vertices of the diamond, we can determine the intercepts:
- When \( x = 0 \), \( |y| = 2 \) gives \( y = 2 \) and \( y = -2 \).
- When \( y = 0 \), \( |x| = 2 \) gives \( x = 2 \) and \( x = -2 \).

Thus, the vertices of the diamond are:
- \( (2, 0) \)
- \( (0, 2) \)
- \( (-2, 0) \)
- \( (0, -2) \)

### Step 3: Sketch the Graph
Sketch the graph using the vertices identified. The diamond will have its corners at the points mentioned above, forming a symmetrical shape centered at the origin.

### Step 4: Calculate the Area of the Diamond
The diamond can be divided into four right triangles, each with a base and height of 2 units:
- The area of one triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 2 = 2 \]

Since there are four identical triangles in the diamond, the total area is:
\[ \text{Total Area} = 4 \times 2 = 8 \]

### Conclusion
The area bounded by the relation \( |x| + |y| = 2 \) is \( 8 \).

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