To find the area of the triangle formed by the points [tex]\((0, -1)\)[/tex], [tex]\((0, 4)\)[/tex], and [tex]\((4, -1)\)[/tex], we can use the determinant formula for the area of a triangle given its vertices [tex]\((x_1, y_1)\)[/tex], [tex]\((x_2, y_2)\)[/tex], and [tex]\((x_3, y_3)\)[/tex]. The formula 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 plug in the given points:
- [tex]\((x_1, y_1) = (0, -1)\)[/tex]
- [tex]\((x_2, y_2) = (0, 4)\)[/tex]
- [tex]\((x_3, y_3) = (4, -1)\)[/tex]
Substitute these values into the formula:
[tex]\[
\text{Area} = \frac{1}{2} \left| 0(4 - (-1)) + 0((-1) - (-1)) + 4((-1) - 4) \right|
\][/tex]
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]
This reduces to:
[tex]\[
\text{Area} = \frac{1}{2} \left| 0 + 0 - 20 \right|
\][/tex]
Simplify the expression inside the absolute value:
[tex]\[
\text{Area} = \frac{1}{2} \left| -20 \right|
\][/tex]
Since the absolute value of -20 is 20, we get:
[tex]\[
\text{Area} = \frac{1}{2} \times 20 = 10
\][/tex]
Therefore, the area of the triangle is [tex]\(10\)[/tex] square units.
The correct answer is:
A. 10 square units
