To find the area of the quadrilateral with vertices [tex]\( P(2t, t) \)[/tex], [tex]\( Q(0, -4) \)[/tex], [tex]\( R(-5, 2) \)[/tex], and [tex]\( S(0, 8) \)[/tex], we can use the Shoelace formula (or Gauss's area formula for polygons). This formula determines the area of a polygon when the coordinates of the vertices are known.
The Shoelace formula for a quadrilateral with vertices [tex]\((x_1, y_1)\)[/tex], [tex]\((x_2, y_2)\)[/tex], [tex]\((x_3, y_3)\)[/tex], and [tex]\((x_4, y_4)\)[/tex] is given by:
[tex]\[
\text{Area} = \frac{1}{2} \left| x_1 y_2 + x_2 y_3 + x_3 y_4 + x_4 y_1 - (y_1 x_2 + y_2 x_3 + y_3 x_4 + y_4 x_1) \right|
\][/tex]
Substituting the coordinates of positions [tex]\( P \)[/tex], [tex]\( Q \)[/tex], [tex]\( R \)[/tex], and [tex]\( S \)[/tex]:
1. [tex]\( P(2t, t) \)[/tex]
2. [tex]\( Q(0, -4) \)[/tex]
3. [tex]\( R(-5, 2) \)[/tex]
4. [tex]\( S(0, 8) \)[/tex]
Using these vertices in the Shoelace formula:
[tex]\[x_1 = 2t, y_1 = t\][/tex]
[tex]\[x_2 = 0, y_2 = -4\][/tex]
[tex]\[x_3 = -5, y_3 = 2\][/tex]
[tex]\[x_4 = 0, y_4 = 8\][/tex]
The Shoelace formula becomes:
[tex]\[
\text{Area} = \frac{1}{2} \left| (2t \cdot -4) + (0 \cdot 2) + (-5 \cdot 8) + (0 \cdot t) - (t \cdot 0) - (-4 \cdot -5) - (2 \cdot 0) - (8 \cdot 2t) \right|
\][/tex]
Simplify each term:
[tex]\[
= \frac{1}{2} \left| -8t + 0 - 40 + 0 - 0 - 20 - 0 - 16t \right|
\][/tex]
Combine like terms inside the absolute value:
[tex]\[
= \frac{1}{2} \left| -8t - 40 - 16t - 20 \right|
\][/tex]
[tex]\[
= \frac{1}{2} \left| -24t - 60 \right|
\][/tex]
Taking the absolute value:
[tex]\[
= \frac{1}{2} \left| 24t + 60 \right|
\][/tex]
[tex]\[
= \frac{1}{2} \cdot 24t + \frac{1}{2} \cdot 60
\][/tex]
[tex]\[
= 12t + 30
\][/tex]
Thus, the area of the quadrilateral [tex]\( PQRS \)[/tex] in terms of [tex]\( t \)[/tex] is:
[tex]\[
\boxed{ \frac{ \left| 24t + 60 \right| }{2} }
\][/tex]
