The radius of the circumcircle of the triangle made by x-axis, y-axis and 4x+3y =12 is

Options

A. 2 unit
B. 2.5 unit
C. 3 unit
D. 4 unit

Answer

To find the radius of the circumcircle of the triangle formed by the x-axis, y-axis, and the line given by the equation \(4x + 3y = 12\), we can follow these steps:

### Step 1: Find the intercepts of the line
To determine the vertices of the triangle, we need to find the x-intercept and y-intercept of the line \(4x + 3y = 12\).

- **Finding the x-intercept**: Set \(y = 0\):
\[ 4x + 3(0) = 12 \implies 4x = 12 \implies x = 3 \]
Thus, the x-intercept is \((3, 0)\).

- **Finding the y-intercept**: Set \(x = 0\):
\[ 4(0) + 3y = 12 \implies 3y = 12 \implies y = 4 \]
Thus, the y-intercept is \((0, 4)\).

### Step 2: Identify the vertices of the triangle
The vertices of the triangle formed by the x-axis, y-axis, and the line are:
- Vertex A: \((0, 0)\) (origin)
- Vertex B: \((3, 0)\) (x-intercept)
- Vertex C: \((0, 4)\) (y-intercept)

### Step 3: Calculate the lengths of the sides of the triangle
We can calculate the lengths of the sides of the triangle using the distance formula.

- **Length of side AB** (between points A and B):
\[ AB = \sqrt{(3 - 0)^2 + (0 - 0)^2} = \sqrt{3^2} = 3 \]

- **Length of side AC** (between points A and C):
\[ AC = \sqrt{(0 - 0)^2 + (4 - 0)^2} = \sqrt{4^2} = 4 \]

- **Length of side BC** (between points B and C):
\[ BC = \sqrt{(3 - 0)^2 + (0 - 4)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Use the circumradius formula for a right triangle
Since triangle ABC is a right triangle (angle at A is \(90^\circ\)), we can use the formula for the circumradius \(R\) of a right triangle:
\[ R = \frac{c}{2} \]
where \(c\) is the length of the hypotenuse (side BC).

From our calculations, \(c = 5\).

### Step 5: Calculate the circumradius
Now we can find the circumradius:
\[ R = \frac{5}{2} = 2.5 \]

### Final Answer
The radius of the circumcircle of the triangle is \(2.5\) units.

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