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Question
Name the quadrilateral formed, if any, by the following points, and given reasons for your answers:
A(-1,-2) B(1, 0), C (-1, 2), D(-3, 0)
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Solution
A (-1,-2) , B(1,0), C(-1,2), D(-3,0)
Let A, B, C and D be the four vertices of the quadrilateral ABCD.
We know the distance between two points `P(x_1, y_1)` and `Q(x_2, y_2)`is given by distance formula:
`PQ = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`
Hence
`=> AB= sqrt((1 - (-1))^2 + (0 - (-2))^2)`
`=> AB= sqrt((2)^2 + (2)^2)`
`=> AB = sqrt(4+ 4)`
`=> AB = 2sqrt2`
Similarly
`=> BC= sqrt((-1)-1)^2 + (2 - 0)^2`
`=> BC= sqrt((-2)^2 + (2)^2)`
`=> BC = sqrt(4 + 4)`
`=> BC= sqrt8`
Similarly,
`=> CD = sqrt((-3)-(-1)^2 + (0 - (2))^2)`
`=> CD = sqrt((-2)^2 + (-2)^2)`
`=> CD = sqrt(4 + 4)`
`=> CD = sqrt8`
`=> CD = 2sqrt2`
Also
`=> DA = sqrt(((-1)-(-3))^2 + (0 -(-2))^2)`
`=> DA = sqrt((2)^2 + (2)^2)`
`=> DA = sqrt(4 + 4)`
`=> DA = sqrt8`
`=> DA = 2sqrt2`
Hence from above we see that all the sides of the quadrilateral are equal. Hence it is a square.
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