Find the area of the triangle which the straight line 3x + 4y - 7 = 0 makes with the co-ordinate axes

Options

A. `(49)/(24)`
B. `(24)/(39)`
C. 2
D. None of these

Answer

To find the area of the triangle formed by the line \(3x + 4y - 7 = 0\) with the coordinate axes, we can follow these steps:

### Step 1: Convert the line equation to intercept form
We start with the equation of the line:
\[ 3x + 4y - 7 = 0 \]
Rearranging this gives:
\[ 3x + 4y = 7 \]

### Step 2: Find the x-intercept
To find the x-intercept, we set \(y = 0\):
\[ 3x + 4(0) = 7 \implies 3x = 7 \implies x = \frac{7}{3} \]
Thus, the x-intercept is \(\left(\frac{7}{3}, 0\right)\).

### Step 3: Find the y-intercept
To find the y-intercept, we set \(x = 0\):
\[ 3(0) + 4y = 7 \implies 4y = 7 \implies y = \frac{7}{4} \]
Thus, the y-intercept is \(\left(0, \frac{7}{4}\right)\).

### Step 4: Calculate the area of the triangle
The area \(A\) of a triangle formed by the x-intercept and y-intercept with the origin can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is the x-intercept \(\frac{7}{3}\) and the height is the y-intercept \(\frac{7}{4}\):
\[ A = \frac{1}{2} \times \frac{7}{3} \times \frac{7}{4} \]

### Step 5: Simplify the area calculation
Calculating the area:
\[ A = \frac{1}{2} \times \frac{7 \times 7}{3 \times 4} = \frac{1}{2} \times \frac{49}{12} = \frac{49}{24} \]

### Final Answer
Thus, the area of the triangle is:
\[ \frac{49}{24} \text{ square units} \]

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