Prove that (0,5), (- 2,- 2), (5,0) and (7,7) are the vertices of a rhombus.

Hint: First draw the quadrilateral by using the given vertices and find the length of each side and diagonals of the quadrilateral. If the length of each side of the quadrilateral are the same then we can say the given points form the vertices of the rhombus.

Complete step by step answer:

Let A(0,5), B(- 2,- 2), C(5,0) and D(7,7) be the vertices of the quadrilateral.

As we know that the length of all sides of a rhombus are equal and the length of its diagonals are not equal.

So, to prove that A,B,C and D are the vertices of a rhombus we have to find the length of its sides and diagonals.

As we know that the length of line joining two points $\left( {{x_1},{y_1}} \right){\text{ and }}\left( {{x_2},{y_2}} \right){\text{ is }}\sqrt {\left( {{{\left( {{x_1} - {x_2}} \right)}^2} + {{\left( {{y_1} - {y_2}} \right)}^2}} \right)} $

$\Rightarrow$ Length of side AB = $\sqrt {\left( {{{\left( {0 - ( - 2)} \right)}^2} + {{\left( {5 - ( - 2)} \right)}^2}} \right)} = \sqrt {53} $

$\Rightarrow$ Length of side BC =$ \sqrt {\left( {{{\left( { - 2 - 5} \right)}^2} + {{\left( { - 2 - 0} \right)}^2}} \right)} = \sqrt {53} $

$ \Rightarrow$ Length of side CD = $\sqrt {\left( {{{\left( {5 - 7} \right)}^2} + {{\left( {0 - 7} \right)}^2}} \right)} = \sqrt {53}$

$\Rightarrow$ Length of side DA = $\sqrt {\left( {{{\left( {7 - 0} \right)}^2} + {{\left( {7 - 5} \right)}^2}} \right)} = \sqrt {53}$

$\Rightarrow$ Length of side AC = $\sqrt {\left( {{{\left( {0 - 5} \right)}^2} + {{\left( {5 - 0} \right)}^2}} \right)} = \sqrt {50}$

$\Rightarrow$ Length of side BD = $\sqrt {\left( {{{\left( { - 2 - 7} \right)}^2} + {{\left( { - 2 - 7} \right)}^2}} \right)} = \sqrt {162}$

As we see here, the length of all the sides of a quadrilateral are equal but the length of diagonals are different.

Hence A(0,5), B(- 2,- 2), C(5,0), D(7,7) are the vertices of rhombus.

NOTE: - Whenever we come up with this type of problem first find the length of all sides and diagonals and if necessary then find the angle between sides. Using properties of quadrilaterals find what type of quadrilateral does the given points form.