Answer:
[tex]A=42\ units^{2}[/tex]
Step-by-step explanation:
see the attached figure with letters to better understand the problem
we know that
The figure ABCD is a rectangle
so
[tex]AB=CD \\AD=BC[/tex]
The area of a rectangle is equal to
[tex]A=B*h[/tex]
where
B is the base
h is the height
the base B is equal to the distance AB
the height h is equal to the distance BC
Let
[tex]A(2,-1),B(9,-8),C(12,-5),D(5,2)[/tex]
Step 1
Find the distance AB
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
[tex]A(2,-1),B(9,-8)[/tex]
substitute the values
[tex]d=\sqrt{(-8+1)^{2}+(9-2)^{2}}[/tex]
[tex]d=\sqrt{(-7)^{2}+(7)^{2}}[/tex]
[tex]dAB=\sqrt{98}=7\sqrt{2}\ units[/tex]
Step 2
Find the distance BC
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
[tex]B(9,-8),C(12,-5)[/tex]
substitute the values
[tex]d=\sqrt{(-5+8)^{2}+(12-9)^{2}}[/tex]
[tex]d=\sqrt{(3)^{2}+(3)^{2}}[/tex]
[tex]dBC=\sqrt{18}=3\sqrt{2}\ units[/tex]
Step 3
Find the area
[tex]A=B*h[/tex]
we have
[tex]B=7\sqrt{2}\ units[/tex]
[tex]h=3\sqrt{2}\ units[/tex]
substitute
[tex]A=7\sqrt{2}*3\sqrt{2}=42\ units^{2}[/tex]
