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Show that points A (4,-1)B (6,0) C (7,-2...

Show that points A (4,-1)B (6,0) C (7,-2) and D (5,-3) are the veties of a square.

Text Solution

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`A(4,-1),B(6,0),C(7,-2)` and `D(5,-3)`.
By distance formula,
`AB=sqrt((6-4)^(2)+[0-(-1)]^(2))`
`=sqrt(2^(2)+1^(2))`
`=sqrt(4+1)`
`=sqrt(5)`….1
`BC=sqrt((7-6)^(2)+(-2-0)^(2))`
`=sqrt((1)^(2)+(-2)^(2))`
`=sqrt(1+4)`
`=sqrt(5)`............2
`CD=sqrt((5-7)^(2)+[-3-(-2)]^(2))`
`=sqrt((-2)^(2)+(-3+2)^(2))`
`=sqrt((-2)^(2)+(-1)^(2))`
`=sqrt(4+1)`
`=sqrt(5)`.........3
`AD=sqrt((5-4)^(2)+[-3-(-1)]^(2))`
`=sqrt((1)^(2)+(-2)^(2))`
`=sqrt(1+4)`
`=sqrt(5)`............4
`:.` from 1, 2, 3 and 4
`AB=BC=CD=AD`
`:.square ABCD` is a rhombus.
By distance formula,
`AC=sqrt((7-4)^(2)+[-2-(-1)]^(2))`
`=sqrt(3^(2)+(-1)^(2))`
`=sqrt(9+1)`
`=sqrt(10)`.........5
`BD=sqrt((5-6)^(2)+(-3-0)^(2))`
`=sqrt((-1)^(2)+(-3)^(2))`
`=sqrt(1+9)`
`=sqrt(10)`........6
In rhombus ABCD
`AC=BD`.[From 5 and 6]
`:.squareABCD` is a square ...(A rhombus is a sdquare if its diagonals are equal)
Points `A(4,-1),B(6,0),C(7,-2)` and `D(5,-3)` are the vertices of a square.
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