**Question:**

Show that the points A(2, -2), B(8, 4), C(5, 7) and D(-1, 1) are the angular points of a rectangle.

**Solution:**

Given: The 4 points are A(2, -2), B(8, 4), C(5, 7) and D(-1, 1).

Note: For a quadrilateral to be a rectangle, the opposite sides of the quadrilateral must be equal and the diagonals must be equal as well.

$A B=\sqrt{36+36}$

$=6 \sqrt{2}$ units $\ldots . .(1)$

$B C=\sqrt{9+9}$

$=3 \sqrt{2}$ units $\ldots(2)$

$C D=\sqrt{36+36}$

$=6 \sqrt{2}$ units ......(3)

$A D=\sqrt{9+9}$ ............(4)

From equations $1,2,3$ and 4, we have

$A B=C D$ and $B C=A D \ldots . .(5)$

Also, $A C=\sqrt{9+81}$

$=3 \sqrt{10}$ units

$B D=\sqrt{81+9}$

$=3 \sqrt{10}$ units

Thus, $A C=B D$.........(6)

From equations 5 and 6, we can conclude that the opposite sides of quadrilateral ABCD are equal and the diagonals of ABCD are equal as well.

Therefore, point $A, B, C$ and $D$ are the angular points of a rectangle.