Step 1: Understanding the Question:
The question asks for the coordinates of the incenter of a triangle with given vertices A, B, and C. The incenter is the point of concurrency of the angle bisectors of a triangle.
Step 2: Key Formula or Approach:
The coordinates of the incenter (I) of a triangle with vertices A(\(x_1, y_1\)), B(\(x_2, y_2\)), and C(\(x_3, y_3\)) are given by the formula:
\[
I = \left( \frac{ax_1 + bx_2 + cx_3}{a+b+c}, \frac{ay_1 + by_2 + cy_3}{a+b+c} \right)
\]
where \(a, b, c\) are the lengths of the sides opposite to vertices A, B, and C, respectively.
We need to calculate the side lengths first.
Step 3: Detailed Explanation:
The given vertices are A(0,0), B(3,0), and C(0,-4).
Let's assign (\(x_1, y_1\)) = (0,0), (\(x_2, y_2\)) = (3,0), (\(x_3, y_3\)) = (0,-4).
Calculate side lengths:
- Side \(a\) is the length of BC (opposite to vertex A):
\[ a = \sqrt{(0-3)^2 + (-4-0)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
- Side \(b\) is the length of AC (opposite to vertex B):
\[ b = \sqrt{(0-0)^2 + (-4-0)^2} = \sqrt{0^2 + (-4)^2} = \sqrt{16} = 4 \]
- Side \(c\) is the length of AB (opposite to vertex C):
\[ c = \sqrt{(3-0)^2 + (0-0)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3 \]
The perimeter is \(a+b+c = 5+4+3 = 12\).
Calculate incenter coordinates:
- x-coordinate of incenter:
\[ I_x = \frac{a x_1 + b x_2 + c x_3}{a+b+c} = \frac{5(0) + 4(3) + 3(0)}{12} = \frac{0 + 12 + 0}{12} = \frac{12}{12} = 1 \]
- y-coordinate of incenter:
\[ I_y = \frac{a y_1 + b y_2 + c y_3}{a+b+c} = \frac{5(0) + 4(0) + 3(-4)}{12} = \frac{0 + 0 - 12}{12} = \frac{-12}{12} = -1 \]
So, the coordinates of the incenter are (1, -1).
Step 4: Final Answer:
The calculated coordinates of the incenter are (1, -1). Comparing this result with the given options, none of the options match. There seems to be an error in the question's options. For the given vertices, the correct incenter is (1, -1).