Question 1204418
<font color=black size=3>
Please post <font size=4><u>one</u></font> problem at a time.


I'll focus on problem 1 only.


I'll reorder the points from
(-2, 3), (5, 8), (3, 3), and (0, 8)
to
(-2, 3), (3, 3), (5, 8), and (0, 8)
and I'll give those points the labels A,B,C,D.


A = (-2,3)
B = (3,3)
C = (5,8)
D = (0,8)


Plot those points on the xy plane. 
{{{
drawing(400,400,-4,7,-2,10,
graph(400,400,-4,7,-2,10),
circle(-2,3,0.05),circle(-2,3,0.07),circle(-2,3,0.09),circle(-2,3,0.11),circle(-2,3,0.13),circle(-2,3,0.15),circle(3,3,0.05),circle(3,3,0.07),circle(3,3,0.09),circle(3,3,0.11),circle(3,3,0.13),circle(3,3,0.15),circle(5,8,0.05),circle(5,8,0.07),circle(5,8,0.09),circle(5,8,0.11),circle(5,8,0.13),circle(5,8,0.15),circle(0,8,0.05),circle(0,8,0.07),circle(0,8,0.09),circle(0,8,0.11),circle(0,8,0.13),circle(0,8,0.15),
line(-2,3,3,3),
line(3,3,5,8),
line(5,8,0,8),
line(0,8,-2,3),
locate(-2-0.5,3-0.2,"A"),locate(3+0.2,3-0.2,"B"),locate(5+0.2,8,"C"),locate(0+0.1,8+0.5,"D")
)
}}}
Because points A and B share the same y coordinate, it indicates segment AB is horizontal.
The horizontal segment allows us to quickly compute the segment length.


Segment AB = 5 units long because it's the distance from -2 to 3 on the number line.
Count out the spaces. Alternatively subtract x coordinates and use absolute value.


AB = 5 is the base.


The vertical distance from A(-2,3) to C(5,8) is 5 units because we subtract y coordinates: 8-3 = 5
This represents the height of the parallelogram.
Keep in mind the base and height are <u>always</u> perpendicular to each other.



Then,
area = base*height
area = 5*5
area = 25 square units


<font color=black size=4>The final answer to problem 1 is <u>25 square units</u>.</font>


-----------------------------------------


Another method


I'll use something called the shoelace formula.
The naming comes from the idea we form a shoelace pattern as shown in the diagram here
<img width=200 src = "https://upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Shoelace3.png/270px-Shoelace3.png">
Image Source
<a href="https://en.wikipedia.org/wiki/Shoelace_formula">https://en.wikipedia.org/wiki/Shoelace_formula</a>

 
Here are alternative names for it:
Gauss's area formula
Surveyor's formula


It appears that tutor @erinonimus is using some variation of this formula.


To set things up, place the coordinates of A through D in four rows like this
A = (-2,3)
B = (3,3)
C = (5,8)
D = (0,8)


Let's delete the labels
(-2,3)
(3,3)
(5,8)
(0,8)


Afterward, we make a copy of the first point (-2,3) and place it at the bottom of the list
(-2,3)
(3,3)
(5,8)
(0,8)
(-2,3)
This will help the list form a loop.



Let's space out the x and y coordinates
{{{
drawing(300,200,-5,5,-5,5,
locate(-3.5714,2.1428,-2),locate(-1.5714,2.1428,3),
locate(-3.5714,0.7142,3),locate(-1.5714,0.7142,3),
locate(-3.5714,-0.7144,5),locate(-1.5714,-0.7144,8),
locate(-3.5714,-2.143,0),locate(-1.5714,-2.143,8),
locate(-3.5714,-3.5716,-2),locate(-1.5714,-3.5716,3),
locate(-3.5714, 3.5714,matrix(1,2,"Shoelace","Formula"))
)
}}}



After that is set up, draw the diagonals shown here
{{{
drawing(300,200,-5,5,-5,5,
locate(-3.5714,2.1428,-2),locate(-1.5714,2.1428,3),
locate(-3.5714,0.7142,3),locate(-1.5714,0.7142,3),
locate(-3.5714,-0.7144,5),locate(-1.5714,-0.7144,8),
locate(-3.5714,-2.143,0),locate(-1.5714,-2.143,8),
locate(-3.5714,-3.5716,-2),locate(-1.5714,-3.5716,3),
red(line(-3.0714,1.4428,-1.7214,0.3642)),
red(line(-3.0714,0.0142,-1.7214,-1.0644)),
red(line(-3.0714,-1.4144,-1.7214,-2.493)),
red(line(-3.0714,-2.843,-1.7214,-3.9216)),
blue(line(-3.0714+0.2,-3.9216,-1.7214,-2.843)),
blue(line(-3.0714,-2.493,-1.7214,-1.4144)),
blue(line(-3.0714,-1.0644,-1.7214,0.0142)),
blue(line(-3.0714,0.3642,-1.7214,1.4428)),
locate(-3.5714, 3.5714,matrix(1,2,"Shoelace","Formula"))
)
}}}
We multiply along the red diagonal pairs and add up those products
<font color=red>-2*3</font>+<font color=red>3*8</font>+<font color=red>5*8</font>+<font color=red>0*3</font> = -6+24+40+0 = <font color=red>58</font>


Do the same for the blue diagonal pairs as well
<font color=blue>-2*8</font>+<font color=blue>0*8</font>+<font color=blue>5*3</font>+<font color=blue>3*3</font> = -16+0+15+9 = <font color=blue>8</font>


Subtract the results
<font color=red>58</font> - <font color=blue>8</font> = 50
If this result was negative, then we'd take the absolute value of it to ensure it's positive.


Lastly, take half of this to get the area = (1/2)*50 = 25


<font color=black size=4>The final answer to problem 1 is <u>25 square units</u>.</font>


A word of caution: 
If you swap the positions of B and C, then the shoelace formula will yield an area of 0. 
It's a strangely fascinating situation where one half of the area cancels out with the other half (it forms a bow-tie shaped figure).


{{{
drawing(400,400,-4,7,-2,10,
graph(400,400,-4,7,-2,10),
circle(-2,3,0.05),circle(-2,3,0.07),circle(-2,3,0.09),circle(-2,3,0.11),circle(-2,3,0.13),circle(-2,3,0.15),circle(3,3,0.05),circle(3,3,0.07),circle(3,3,0.09),circle(3,3,0.11),circle(3,3,0.13),circle(3,3,0.15),circle(5,8,0.05),circle(5,8,0.07),circle(5,8,0.09),circle(5,8,0.11),circle(5,8,0.13),circle(5,8,0.15),circle(0,8,0.05),circle(0,8,0.07),circle(0,8,0.09),circle(0,8,0.11),circle(0,8,0.13),circle(0,8,0.15),
line(-2,3,3,3),
line(3,3,5,8),
line(5,8,0,8),
line(0,8,-2,3),
locate(-2-0.5,3-0.2,"A"),locate(3+0.2,3-0.2,"B"),locate(5+0.2,8,"C"),locate(0+0.1,8+0.5,"D")
)
}}}
vs
{{{
drawing(400,400,-4,7,-2,10,
graph(400,400,-4,7,-2,10),
circle(-2,3,0.05),circle(-2,3,0.07),circle(-2,3,0.09),circle(-2,3,0.11),circle(-2,3,0.13),circle(-2,3,0.15),circle(3,3,0.05),circle(3,3,0.07),circle(3,3,0.09),circle(3,3,0.11),circle(3,3,0.13),circle(3,3,0.15),circle(5,8,0.05),circle(5,8,0.07),circle(5,8,0.09),circle(5,8,0.11),circle(5,8,0.13),circle(5,8,0.15),circle(0,8,0.05),circle(0,8,0.07),circle(0,8,0.09),circle(0,8,0.11),circle(0,8,0.13),circle(0,8,0.15),
line(-2,3,5,8),line(5,8,3,3),line(3,3,0,8),line(0,8,-2,3),
locate(-2-0.5,3-0.2,"A"),locate(3+0.2,3-0.2,"C"),locate(5+0.2,8,"B"),locate(0+0.1,8+0.5,"D")
)
}}}


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Some more practice
<a href="https://www.algebra.com/algebra/homework/Points-lines-and-rays/Points-lines-and-rays.faq.question.1197252.html">https://www.algebra.com/algebra/homework/Points-lines-and-rays/Points-lines-and-rays.faq.question.1197252.html</a>
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