The length of the portion of the straight line ` 3 x + 4y = 12 ` intercepted between the axes is

Options

A. 5
B. 3
C. 4
D. 7

Answer

To find the length of the portion of the straight line \(3x + 4y = 12\) intercepted between the axes, we will follow these steps:

### Step 1: Find the x-intercept
To find the x-intercept, we set \(y = 0\) in the equation of the line.

\[ 3x + 4(0) = 12 \implies 3x = 12 \implies x = \frac{12}{3} = 4 \]

So, the x-intercept is \(4\).

### Step 2: Find the y-intercept
To find the y-intercept, we set \(x = 0\) in the equation of the line.

\[ 3(0) + 4y = 12 \implies 4y = 12 \implies y = \frac{12}{4} = 3 \]

So, the y-intercept is \(3\).

### Step 3: Determine the length of the intercepted line segment
The intercepts we found are \(4\) (x-intercept) and \(3\) (y-intercept). These intercepts form a right triangle with the axes. The length of the line segment intercepted between the axes can be found using the Pythagorean theorem.

Let the length of the intercepted segment be \(L\). According to the Pythagorean theorem:

\[ L = \sqrt{(x\text{-intercept})^2 + (y\text{-intercept})^2} \]

Substituting the values we found:

\[ L = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Conclusion
The length of the portion of the straight line \(3x + 4y = 12\) intercepted between the axes is \(5\) units.

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