Answer
Quadrilateral ABCD is a trapezoid
Area of quadrilateral ABCD = 9
Work Step by Step
ABCD is a trapezoid because it has two bases that are not congruent to each other and 2 sides that are congruent
Area of trapezoid = $ \frac{a+b}{2} \times height $
a and b are the two bases ^
base AB = $D =\sqrt (x_{2}-x_{1})^2 + (y_{2}-y_{1})^2$
$D =\sqrt (1-5)^2 + (0-0)^2$
$D =\sqrt (-4)^2 + (0)^2$
$D =\sqrt (16 + 0)$
$D =\sqrt 16$
$D =4$
therefore segment $AB =4$
base DC = $D =\sqrt (x_{2}-x_{1})^2 + (y_{2}-y_{1})^2$
$D =\sqrt (4-2)^2 + (3-3)^2$
$D =\sqrt (2)^2 + (0)^2$
$D =\sqrt (4 + 0)$
$D =\sqrt 4$
$D =2$
therefore segment DC = 2
to find the height you do find the distance from b to point (4,0) and the distance from B to C or the distance from A to D and the distance from point (2,3) to point A where both final 2 distances are equal to each other.
CB = $D =\sqrt (x_{2}-x_{1})^2 + (y_{2}-y_{1})^2$
$D =\sqrt (5-4)^2 + (0-3)^2$
$D =\sqrt (1)^2 + (-3)^2$
$D =\sqrt (1+9)$
$D =\sqrt 10$
therefore segment CB = $\sqrt 10$
$height^2 +1^2 = \sqrt 10^2$
$height^2 +1 = 10$
$height^2 = 9$
height = 3
Area of trapezoid = $ \frac{a+b}{2} \times height $
Area of trapezoid = $ \frac{4+2}{2} \times 3 $
Area of trapezoid = $ \frac{6}{2} \times 3 $
Area of trapezoid = $ 3 \times 3 $
Area of trapezoid = 9