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Question
Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:
(- 1, - 2), (1, 0), (- 1, 2), (- 3, 0)
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Solution
Let the points (−1, −2), (1, 0), (−1, 2), and (−3, 0) represent the vertices A, B, C, and D of the given quadrilateral, respectively.
AB = `sqrt((-1-1)^2+(-2-0)^2)`
= `sqrt((-2)^2+(-2)^2)`
= `sqrt(4+4)`
= `sqrt8`
=`2sqrt2`
BC = `sqrt((1-(-1))^2+(0-2)^2)`
= `sqrt((2)^2+(-2)^2)`
= `sqrt(4+4)`
= `sqrt8`
= `2sqrt2`
CB = `sqrt((-1-(-3))^2+(2-0)^2) `
= `sqrt((2)^2+(2)^2)`
= `sqrt(4+4)`
= `sqrt8 `
= `2sqrt2`
AD = `sqrt((-1-(3))^2 + (-2-0)^2)`
= `sqrt((2)^2+(-2)^2)`
= `sqrt(4+4)`
= `sqrt8`
= `2sqrt2`
Diagonal AC = `sqrt((-1-(-1))^2+(-2-2)^2)`
= `sqrt(0^2+(-4)^2)`
= `sqrt(16) `
= 4
Diagonal BD = `sqrt((1-(-3))^2+(0-0)^2)`
= `sqrt((4)^2+0^2)`
= `sqrt16 `
= 4
It can be observed that all sides of this quadrilateral are of the same length and also, the diagonals are of the same length. Therefore, the given points are the vertices of a square.
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