To find the area of the triangle with vertices [tex]\((-8, -1)\)[/tex], [tex]\((3, -8)\)[/tex], and [tex]\((3, -1)\)[/tex], follow these steps:
1. Identify the vertices: The given vertices of the triangle are:
- [tex]\( A(-8, -1) \)[/tex]
- [tex]\( B(3, -8) \)[/tex]
- [tex]\( C(3, -1) \)[/tex]
2. Calculate the area using the vertex coordinates: The area of a triangle with coordinates [tex]\((x_1, y_1)\)[/tex], [tex]\((x_2, y_2)\)[/tex], [tex]\((x_3, y_3)\)[/tex] can be determined using the formula:
[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]
3. Substitute the given coordinates into the formula:
[tex]\[
x_1 = -8, \quad y_1 = -1, \quad x_2 = 3, \quad y_2 = -8, \quad x_3 = 3, \quad y_3 = -1
\][/tex]
4. Evaluate each term inside the absolute value:
- Calculate [tex]\( x_1(y_2 - y_3) \)[/tex]:
[tex]\[
-8 \times (-8 - (-1)) = -8 \times (-8 + 1) = -8 \times -7 = 56
\][/tex]
- Calculate [tex]\( x_2(y_3 - y_1) \)[/tex]:
[tex]\[
3 \times (-1 - (-1)) = 3 \times (-1 + 1) = 3 \times 0 = 0
\][/tex]
- Calculate [tex]\( x_3(y_1 - y_2) \)[/tex]:
[tex]\[
3 \times (-1 - (-8)) = 3 \times (-1 + 8) = 3 \times 7 = 21
\][/tex]
5. Sum these terms inside the absolute value:
[tex]\[
56 + 0 + 21 = 77
\][/tex]
6. Multiply by [tex]\( \frac{1}{2} \)[/tex] and take the absolute value:
[tex]\[
\text{Area} = \frac{1}{2} \left| 77 \right| = \frac{1}{2} \times 77 = 38.5
\][/tex]
Thus, the area of the triangle with vertices [tex]\((-8, -1)\)[/tex], [tex]\((3, -8)\)[/tex], and [tex]\((3, -1)\)[/tex] is [tex]\( 38.5 \)[/tex] square units.
