Area of triangle fomed by the straight line `8x - 3y = 24` and coordinate axes is

Options

A. 24 sq. unit
B. 12 sq. unit
C. 6 sq. unit
D. 18 sq. unit

Answer

To find the area of the triangle formed by the line \(8x - 3y = 24\) and the coordinate axes, we can follow these steps:

### Step 1: Find the intercepts of the line

To find the area of the triangle, we first need to determine where the line intersects the x-axis and y-axis.

**Finding the x-intercept:**
Set \(y = 0\) in the equation \(8x - 3y = 24\):
\[ 8x - 3(0) = 24 \implies 8x = 24 \implies x = \frac{24}{8} = 3 \]
So, the x-intercept is at the point \((3, 0)\).

**Finding the y-intercept:**
Set \(x = 0\) in the equation \(8x - 3y = 24\):
\[ 8(0) - 3y = 24 \implies -3y = 24 \implies y = \frac{24}{-3} = -8 \]
So, the y-intercept is at the point \((0, -8)\).

### Step 2: Plot the points and form the triangle

We have the intercepts:
- x-intercept: \((3, 0)\)
- y-intercept: \((0, -8)\)

Now, we can visualize the triangle formed by these points and the origin \((0, 0)\). The triangle has vertices at \((0, 0)\), \((3, 0)\), and \((0, -8)\).

### Step 3: Calculate the area of the triangle

The area \(A\) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In our case:
- The base of the triangle is the distance along the x-axis from \((0, 0)\) to \((3, 0)\), which is \(3\) units.
- The height of the triangle is the distance along the y-axis from \((0, 0)\) to \((0, -8)\), which is \(8\) units (we take the absolute value since area is always positive).

Substituting these values into the area formula:
\[ A = \frac{1}{2} \times 3 \times 8 = \frac{24}{2} = 12 \text{ square units} \]

### Final Answer:
The area of the triangle formed by the line \(8x - 3y = 24\) and the coordinate axes is \(12\) square units.

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