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Question
Name the type of quadrilateral formed, if any, by the following point, and give reasons for your answer:
(4, 5), (7, 6), (4, 3), (1, 2)
Solution
Let the points (4, 5), (7, 6), (4, 3), and (1, 2) be representing the vertices A, B, C, and D of the given quadrilateral respectively.
∴ AB = `sqrt((4-7)^2+(5-6)^2)`
= `sqrt((-3)^2+(-1)^2)`
= `sqrt(9+1)`
= `sqrt10`
BC = `sqrt((7-4)^2+(6-3)^2)`
= `sqrt((3)^2+(3)^2)`
= `sqrt(9+9)`
= `sqrt18`
CD = `sqrt((4-1)^2+(3-2)^2)`
= `sqrt((3)^2+(1)^2)`
= `sqrt(9+1)`
= `sqrt10`
AD = `sqrt((4-1)^2+(5-2)^2)`
= `sqrt((3)^2+(3)^2)`
= `sqrt(9+9)`
= `sqrt18`
Diagonal AC = `sqrt((4-4)^2+(5-3)^2)`
= `sqrt((0)^2+(2)^2)`
= `sqrt(0+4)`
= 2
Diagonal CD = `sqrt((7-1)^2 + (6-2)^2)`
= `sqrt((6)^2+(4)^2)`
= `sqrt(36+16)`
= `sqrt52`
= `13sqrt2`
It can be observed that opposite sides of this quadrilateral are of the same length. However, the diagonals are of different lengths. Therefore, the given points are the vertices of a parallelogram.
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