State True or False that (0,5),(-2,-2),(5,0),(7,7)are the vertices of a rhombus.

# State True or False that $\left(0,5\right),\left(-2,-2\right),\left(5,0\right),\left(7,7\right)$are the vertices of a rhombus.

1. A
True
2. B
False

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### Solution:

Let us consider as the vertices of the quadrilateral.
As we know that the length of all sides of a rhombus are equal whereas the length of the diagonal of the rhombus are not equal.
So, to prove that A,B,C and D are the vertices of a rhombus it is necessary to find the length of its sides and diagonals.
The length of line that joins two points

Length of the side AB =  $\sqrt{{\left({0}{-}\left({-}{2}\right)\right)}^{{2}}{+}{\left({5}{-}\left({-}{2}\right)\right)}^{{2}}{=}}\sqrt{{53}}$
Length of the side BC= $\sqrt{{\left({-}{2}{-}{5}\right)}^{{2}}{+}{\left({-}{2}{-}{0}\right)}^{{2}}}$  ${=}$ $\sqrt{{53}}$
Length of the side CD =
Length of the side AD= $\sqrt{{\left({7}{-}{0}\right)}^{{2}}{+}{\left({7}{-}{5}\right)}^{{2}}}{=}\sqrt{{53}}$
Length of the side AC = $\sqrt{{\left({0}{-}{5}\right)}^{{2}}{+}{\left({5}{-}{0}\right)}^{{2}}}{=}\sqrt{{50}}$
Length of the side BD =
Therefore, (0 , 5), (- 2,- 2), (5,0) and (7,7) are the vertices of a rhombus as the length of each side of the rhombus is equal and it has unequal diagonals.

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