Answer with explanation:
The Vertices of quadrilateral A B CD, are A(1, a - 3), B(10, a), C(9, a + 3) and D(0, a).
Mid point of AC= (5,1)
Mid point of B D= (5,1)
Mid point formula of two points having coordinates , (a,b) and (c,d) is , if (x,y), then
[tex]x=\frac{a+b}{2},y=\frac{c+d}{2}[/tex]
[tex]So,\rightarrow \frac{a-3+a+3}{2}=1\\\\\rightarrow \frac{2a}{2}=1\\\\\rightarrow a=1[/tex]
So, the coordinates of vertices of quadrilateral A B CD, are A(1, -2), B(10, 1), C(9, 4) and D(0, 1).
Distance formula of two points having coordinates, (a,b) and (c,d) is,
[tex]=\sqrt{(a-c)^2+(b-d)^2}[/tex]
[tex]AB=\sqrt{(10-1)^2+(1+2)^2}=\sqrt{81+9}=\sqrt{90}=3\sqrt{10}\\\\BC=\sqrt{(10-9)^2+(1-4)^2}\\\\=\sqrt{1+9}\\\\=\sqrt{10}\\\\CD=\sqrt{(9-0)^2+(3-0)^2}=\sqrt{90}=3\sqrt{10}\\\\DA=\sqrt{(1-0)^2+(1+2)^2}=\sqrt{10}\\\\AC=\sqrt{(9-1)^2+(4+2)^2}=\sqrt{64+36}\\\\AC=\sqrt{100}\\\\AC=10\\\\BD=\sqrt{(10-0)^2+(1-1)^2}\\\\BD=10[/tex]
Opposite sides AB and CD are equal to 3√10 unit and , BC and AD are equal to √10 unit.
Also,length of Diagonals , AC=B D=10 unit.
∴ The Quadrilateral, AB CD is a Rectangle.
