Find the length of intercept b/w both axis by a line `3x+4y=12`

Answer

To find the length of the intercept between the x-axis and y-axis for the line given by the equation \(3x + 4y = 12\), we can follow these steps:

### Step 1: Rewrite the equation in intercept form
We start with the equation of the line:
\[ 3x + 4y = 12 \]
To express this in intercept form, we divide the entire equation by 12:
\[ \frac{3x}{12} + \frac{4y}{12} = 1 \]
This simplifies to:
\[ \frac{x}{4} + \frac{y}{3} = 1 \]

### Step 2: Identify the intercepts
From the equation \(\frac{x}{4} + \frac{y}{3} = 1\), we can identify the x-intercept and y-intercept:
- The x-intercept occurs when \(y = 0\):
\[ \frac{x}{4} + 0 = 1 \implies x = 4 \]
- The y-intercept occurs when \(x = 0\):
\[ 0 + \frac{y}{3} = 1 \implies y = 3 \]

### Step 3: Calculate the length of the intercept
The intercepts on the axes are (4, 0) and (0, 3). We can visualize this as a right triangle formed by these intercepts, where one leg is the x-intercept (4 units) and the other leg is the y-intercept (3 units).

### Step 4: Apply the Pythagorean theorem
To find the length of the intercept (the hypotenuse of the triangle), we apply the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
where \(a = 4\) (x-intercept) and \(b = 3\) (y-intercept):
\[ c^2 = 4^2 + 3^2 = 16 + 9 = 25 \]
Taking the square root gives:
\[ c = \sqrt{25} = 5 \]

### Conclusion
The length of the intercept between the x-axis and y-axis for the line \(3x + 4y = 12\) is \(5\) units.

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