We want to find the area of the polygon
joining the points A, B, C, D, and E.
Notice how we have five points,
which I've already plotted on the coordinate plane.
By connecting the points we can see we have a pentagon,
which is a five sided polygon.
So, our goal is to find the area of the pentagon.
We might be thinking to find the area
we should break up the inside into triangles,
maybe here and here,
but to find the area of these triangles,
we'd have to find the base and the height of each of them,
since the area of a triangle
is equal to 1/2 base times height.
It would be fairly challenging to find
the base and the height for each of these triangles,
so I think it'll be easier if we think outside the box,
or in this case, think outside the pentagon.
What we'll do is construct a rectangle
where the sides are parallel to the x and y axes,
and pass through as many of the vertices
of the pentagon as possible.
So, for example, if we start on the left,
this side must be parallel to the y-axis
and pass through point E, go as high as point A,
and as low as point C.
And therefore, the left side would look like this.
The right side must pass through point B
and be parallel to the other side,
as well as go as high as point A, and as low as point C,
and therefore, the right side would look like this.
And now, we finish off the rectangle
passing through point A on top,
and passing through point C on the bottom.
So, notice how we have one, two, three right triangles
outside the pentagon, and over here, we have an odd shape.
What we can do, though, is cut this
into a trapezoid and a triangle by sketching a segment here.
So, now what we'll do is find the area
of the large rectangle,
and then subtract out the extra area,
which will consist of one, two, three, four right triangles,
and a trapezoid.
Let's begin by determining the dimensions of the rectangle.
Notice along the x-axis, the left side is at negative three,
the right side is at positive eight.
The length of the side is 11,
which we can also count if we need to.
And then, this area on the right,
which is parallel to the y-axis,
notice how the y values go from negative four
to positive seven, and therefore, the length is also 11.
So, we actually have a largest square
where the pentagon is inside the square.
So, again, our strategy is going to be to find the area
of the large square, and then subtract out the area
of the four right triangles and the trapezoid,
which should be the areas shaded here in red.
Let's write out our formula.
The area of the pentagon, which I'll call A sub P
is equal to the area of the large square,
which we'll call A sub S,
and then minus the sum of the areas in red,
which will be the areas of four right triangles
and a trapezoid.
So, we'll call the areas of the triangles A sub T sub one
through A sub T sub four.
So, we have A sub T sub one,
plus A sub T sub two, plus A sub T sub three,
plus A sub T sub four.
And then, for the area of the trapezoid,
let's use A sub TR.
Before we determine the area in red,
let's go ahead and label the dimensions that we need.
Because we have right triangles,
we can use the two legs as the base and the height.
Starting in the upper left-hand corner
with this larger right triangle,
one leg has a length of one, two, three, four, five units.
The longer leg has a length of eight units.
Moving clockwise, the short leg of this right triangle
has a length of three units.
The longer leg has a length of eight units.
The small right triangle here
in the lower right-hand corner,
one leg has a length of two units.
The other leg has a length of three units.
And then, for the right triangle on the bottom here,
the short leg has a length of two units.
The longer leg has a length of seven units.
And then for the trapezoid,
the bases are the two parallel sides.
We already know one base has a length of two units.
The longer base has a length of six units,
and the height will be this length here, which is two units.
Now that we have all the dimensions we need,
let's find the area of the pentagon.
The area of the pentagon is equal to the area of the square,
which is 11 times 11.
And then, minus the sum of the areas shaded in red,
which consists of four right triangles and a trapezoid.
So, again, starting in the upper left-hand corner
and moving clockwise, the area of this larger triangle
is equal to 1/2 five times eight.
Plus moving clockwise, the area of the triangle
in the upper right-hand corner
is 1/2 times three times eight.
Plus the small right triangle in the lower right-hand corner
has an area of 1/2 times two times three.
The next right triangle has an area
of 1/2 times two times seven.
And finally, the area of the trapezoid
is equal to 1/2 times the sum of the two bases,
which will give us six plus two
times the height, which is two units.
And now, let's begin simplifying.
The area of the pentagon is equal to 11 times 11 is 121,
and then minus the sum of these products.
1/2 times five times eight equals 1/2 times 40, which is 20.
Plus 1/2 times three times eight
is equal to 1/2 times 24, or 12.
Plus 1/2 times two times three
is equal to 1/2 times six, which is three.
Plus 1/2 times two times seven,
which is 1/2 times 14, or seven.
And then, plus 1/2 times the sum of six and two times two.
Well, six plus two is eight, eight times two is 16.
1/2 times 16 is eight.
This gives us 121 minus 20 plus 12
plus three plus seven plus eight is equal to 50.
121 minus 50 is equal to 71.
The area of the pentagon is equal to 71 square units,
which is the light blue area shaded here.
The area inside the pentagon
enclosed by points, A, B, C, D, and E.
I hope you found this helpful.