First let's calculate the area of ABCD.
Using AD as the base, the height can be calculated with the difference of y-coordinates of the points B and A, and the base is the difference of x-coordinates of the points D and A:
[tex]\begin{gathered} h=2b-0=2b \\ base=2c-0=2c \end{gathered}[/tex]So the area of ABCD is:
[tex]A=2b\cdot2c=4bc[/tex]Now, finding the coordinates of the midpoints E, F, G and H, we have:
[tex]\begin{gathered} E_x=\frac{2a+0}{2}=a \\ E_y=\frac{2b+0}{2}=b \\ \\ F_x=\frac{2a+2c+2a}{2}=2a+c \\ F_y=\frac{2b+2b}{2}=2b \\ \\ G_x=\frac{2a+2c+2c}{2}=a+2c \\ G_y=\frac{2b+0}{2}=b \\ \\ H_x=\frac{0+2c}{2}=c \\ H_y=\frac{0+0}{2}=0 \end{gathered}[/tex]Using EG as a base of the triangles EGF and EGH, we can calculate the areas of these triangles.
The height of these triangles are the difference in y-coordinates of the point F and E:
[tex]\begin{gathered} h=2b-b=b \\ \text{base}=a+2c-a=2c_{} \\ \text{Area}=\frac{2c\cdot b}{2}=bc \end{gathered}[/tex]Adding the area of the two triangles, we have the area of EFGH:
[tex]A=bc+bc=2bc[/tex]So comparing the area of EFGH and ABCD, we have:
[tex]\frac{2bc}{4bc}=\frac{1}{2}[/tex]Therefore the area of EFGH is half the area of ABCD.
