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Question

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


A
True
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B
False
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Solution

The correct option is A True

Let $$A(1,7), B(4,2), C(-1,-1)$$ 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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