The graph of `3 x + 4y - 24 = 0 ` forms a triangle OAB with the coordinate axes, where O is the origin . Also the graph of ` x + y +4 = 0 ` forms a tringle OCD with the coordinate axes. Then the area of `Delta OCD` is equal to

Options

A. `(1)/(2)` of area of `Delta OAB`
B. `(1)/(3)` of area of `Delta OAB`
C. `(2)/(3)` of area of `Delta OAB`
D. the area of ` Delta OAB`

Answer

To find the area of triangle OCD formed by the line \(x + y + 4 = 0\) with the coordinate axes, we will follow these steps:

### Step 1: Find the intercepts of the line \(3x + 4y - 24 = 0\)

1. **Set \(x = 0\)** to find the y-intercept:
\[ 3(0) + 4y - 24 = 0 \implies 4y = 24 \implies y = 6 \]
So, the y-intercept is \( (0, 6) \).

2. **Set \(y = 0\)** to find the x-intercept:
\[ 3x + 4(0) - 24 = 0 \implies 3x = 24 \implies x = 8 \]
So, the x-intercept is \( (8, 0) \).

### Step 2: Calculate the area of triangle OAB

The area \(A\) of triangle OAB can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is the x-intercept (8) and the height is the y-intercept (6):
\[ A = \frac{1}{2} \times 8 \times 6 = \frac{48}{2} = 24 \text{ square units} \]

### Step 3: Find the intercepts of the line \(x + y + 4 = 0\)

1. **Set \(x = 0\)** to find the y-intercept:
\[ 0 + y + 4 = 0 \implies y = -4 \]
So, the y-intercept is \( (0, -4) \).

2. **Set \(y = 0\)** to find the x-intercept:
\[ x + 0 + 4 = 0 \implies x = -4 \]
So, the x-intercept is \( (-4, 0) \).

### Step 4: Calculate the area of triangle OCD

The area \(A_{OCD}\) of triangle OCD can be calculated similarly:
\[ A_{OCD} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is the distance from the origin to the x-intercept (-4), which is 4, and the height is the distance from the origin to the y-intercept (-4), which is also 4:
\[ A_{OCD} = \frac{1}{2} \times 4 \times 4 = \frac{16}{2} = 8 \text{ square units} \]

### Conclusion

The area of triangle OCD is \(8\) square units.

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