Question

# Say true or false.Points $$(1, 7), (4, 2), (-1, -1)$$ and $$(-4, 4)$$ are the vertices of a square.

A
True
B
False

Solution

## The correct option is A TrueLet $$A(1,7), B(4,2), C(-1,-1)$$ and $$D(-4, 4)$$ be the given co-ordinates.1.  Diagonals $$BD$$ and $$AC$$ BD $$= \sqrt {(4+4)^2+(2-4)^2} = \sqrt {68}$$AC $$=\sqrt {(1+1)^2+(7+1)^2} = \sqrt {68}$$$$\therefore AC=BD$$$$\therefore$$ the diagonals are equal.2. Sides $$AB, BC, CD$$ and $$DA$$ Length $$AB=\sqrt { { \left( {4}{ { 1 } } \right) }^{ 2 }+{ \left( { 2 }{ 7 }\right) }^{ 2 } } =\sqrt { 34 }$$Length  $$BC=\sqrt { { \left( {1}{ { 4 } }\right) }^{ 2 }+{ \left( { 1 }{ 2}\right) }^{ 2 } } =\sqrt { 34 }$$Length $$CD=\sqrt { { \left( {4}{ { 1 } }\right) }^{ 2 }+{ \left( { 4 }{ 1}\right) }^{ 2 } } =\sqrt{ 34 }$$Length $$DA=\sqrt { { \left( {4}{ { 1 } }\right) }^{ 2 }+{ \left( { 4 }{ 7}\right) }^{ 2 } } =\sqrt { 34 }$$Therefore, $$\Box ABCD$$ is a square since sides $$AB, BC, CD$$ and $$DA$$ are equal and the diagonals are also equal.Mathematics

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