Rotation and area

First, it is necessary to work out where B is.

Since A is 4 up and 4 along from the origin, OA makes a $45^\circ$ angle with the $x$ axis. This means that the obtuse angle between OA and the $x$ axis must be $135^\circ$, so B must lie on the $x$ axis. This is shown in the diagrams below, where the red triangle is the triangle whose area we need to find.

The distances OA and OB are equal, since OB is just a rotation of OA. This distance can be found by considering the triangle below:

Image

 

 This triangle is right-angled and OA is the hypotenuse, so, by Pythagoras' Theorem,

${\text{OA}}^2=4^2+4^2=2\times4^2$ or $32$,

So $\text{OA}=\sqrt{32}=5.657$ or $4\sqrt2$.

Now, there are various different possible methods.

Using the area of a triangle formula

Area = $\frac12$ base x height

         = $\frac12$ OB x $y$ coordinate of A 

         = $\frac12$ x $4\sqrt2$ x 4 

         = $8\sqrt2$ 

 

Using horizontal and vertical triangles and rectangles

Image

 

The yellow and blue triangles are drawn to make a rectangle with the red triangle. The length of the rectangle is OB plus the $x$ coordinate of A, so it is $4+4\sqrt2$ or $4+5.66=9.657$

The height of the rectangle is the $y$ coordinate of A, which is $4$.

So the area of the rectangle is $4(4+4\sqrt2)=16+16\sqrt2$ square units, or $4\times9.657=38.63$ square units.

The area of the yellow triangle is half of the area of the rectangle, so the area of the red and blue triangles combined is also equal to half the area of the rectangle. That area is $(16+16\sqrt2)\div2=8+8\sqrt2$, or $38.63\div2=19.31$, square units.

The blue triangle has base $4$ and heght $4$, so its area is $\frac124\times4=8$ square units.

So the area of the red triangle is $8+8\sqrt2-8=8\sqrt2$, or $19.31-8=11.31$, square units.

Using trigonometry

Image

 This triangle is perfect for the formula $\text{Area}=\frac12ab\sin{C}$, where:

$a=b=4\sqrt2$ or $5.657$, and $C=135^\circ$

So $\text{Area}=\frac12(4\sqrt2)^2\sin{135}=8\sqrt2$ square units.

Cutting up the triangle and rearranging the pieces

Image

 

The triangle is isosceles, so the angles at A and B are equal. Since all of the angles add up to $180^\circ$, these angles must add up to $45^\circ$, or must be $22.5^\circ$ each.

Now imagine cutting the triangle in half along the dotted line, and then rotating the blue part, so that the two