To find the area of the triangle with vertices at [tex]\((0, -1)\)[/tex], [tex]\((0, 4)\)[/tex], and [tex]\((4, -1)\)[/tex], we can use the formula for the area of a triangle when the coordinates of its vertices are known.
The formula for the area of a triangle given vertices [tex]\((x_1, y_1)\)[/tex], [tex]\((x_2, y_2)\)[/tex], and [tex]\((x_3, y_3)\)[/tex] is:
[tex]\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\][/tex]
Let's denote the vertices of our triangle as follows:
- [tex]\(A = (0, -1)\)[/tex]
- [tex]\(B = (0, 4)\)[/tex]
- [tex]\(C = (4, -1)\)[/tex]
Plugging in the coordinates into the formula:
[tex]\[
\text{Area} = \frac{1}{2} \left| 0 \cdot (4 - (-1)) + 0 \cdot ((-1) - (-1)) + 4 \cdot ((-1) - 4) \right|
\][/tex]
Now let's simplify each term inside the absolute value:
[tex]\[
\text{Area} = \frac{1}{2} \left| 0 \cdot 5 + 0 \cdot 0 + 4 \cdot (-5) \right|
\][/tex]
[tex]\[
\text{Area} = \frac{1}{2} \left| 0 + 0 + (-20) \right|
\][/tex]
[tex]\[
\text{Area} = \frac{1}{2} \left| -20 \right|
\][/tex]
[tex]\[
\text{Area} = \frac{1}{2} \cdot 20 = 10
\][/tex]
So, the area of the triangle is [tex]\(10\)[/tex] square units.
Therefore, the correct answer is:
D. 10 square units
