The length of the equation `4x+3y-12=0` that intersects two coordinate axes is

Options

A. 2.5 units
B. 7 units
C. 5 units
D. 6 units

Answer

To find the length of the line segment that intersects the coordinate axes given by the equation \(4x + 3y - 12 = 0\), we can follow these steps:

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

\[ 4x + 3(0) - 12 = 0 \]

This simplifies to:

\[ 4x - 12 = 0 \]

Solving for \(x\):

\[ 4x = 12 \implies x = \frac{12}{4} = 3 \]

Thus, the x-intercept is at the point \(A(3, 0)\).

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

\[ 4(0) + 3y - 12 = 0 \]

This simplifies to:

\[ 3y - 12 = 0 \]

Solving for \(y\):

\[ 3y = 12 \implies y = \frac{12}{3} = 4 \]

Thus, the y-intercept is at the point \(B(0, 4)\).

### Step 3: Calculate the distance between the intercepts
Now, we need to calculate the distance between points \(A(3, 0)\) and \(B(0, 4)\). The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of points \(A\) and \(B\):

\[ d = \sqrt{(0 - 3)^2 + (4 - 0)^2} \]

Calculating this gives:

\[ d = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Final Answer
The length of the line segment that intersects the two coordinate axes is \(5\) units.

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