To determine if the points [tex]\((1, 5)\)[/tex], [tex]\((2, 3)\)[/tex], and [tex]\((-2, -11)\)[/tex] are collinear, we need to find the area of the triangle formed by these three points. If the area of the triangle is zero, then the points are collinear.
We will use the determinant method to calculate the area of the triangle formed by the points [tex]\((x1, y1) = (1, 5)\)[/tex], [tex]\((x2, y2) = (2, 3)\)[/tex], and [tex]\((x3, y3) = (-2, -11)\)[/tex].
The formula for the area of a triangle given three vertices [tex]\((x1, y1)\)[/tex], [tex]\((x2, y2)\)[/tex], and [tex]\((x3, y3)\)[/tex] is:
[tex]\[
\text{Area} = \frac{1}{2} \left| x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) \right|
\][/tex]
Plugging the coordinates into the formula:
[tex]\[
\text{Area} = \frac{1}{2} \left| 1(3 - (-11)) + 2((-11) - 5) + (-2)(5 - 3) \right|
\][/tex]
Simplifying the expression inside the absolute value:
[tex]\[
\text{Area} = \frac{1}{2} \left| 1(14) + 2(-16) + (-2)(2) \right|
\][/tex]
[tex]\[
\text{Area} = \frac{1}{2} \left| 14 + (-32) + (-4) \right|
\][/tex]
[tex]\[
\text{Area} = \frac{1}{2} \left| 14 - 32 - 4 \right|
\][/tex]
[tex]\[
\text{Area} = \frac{1}{2} \left| 14 - 36 \right|
\][/tex]
[tex]\[
\text{Area} = \frac{1}{2} \left| -22 \right|
\][/tex]
[tex]\[
\text{Area} = \frac{1}{2} \times 22
\][/tex]
[tex]\[
\text{Area} = 11
\][/tex]
Since the area of the triangle is [tex]\(11\)[/tex], we can conclude that the points are not collinear. If the points were collinear, the area of the triangle would be zero. Therefore, the points [tex]\((1, 5)\)[/tex], [tex]\((2, 3)\)[/tex], and [tex]\((-2, -11)\)[/tex] are not collinear.
