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Question
Show that the points A (5, 6), B (1, 5), C (2, 1) and D (6, 2) are the vertices of a square ABCD.
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Solution
We use the distance formula = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)` to find the length of each side.
AB = `sqrt((1 - 5)^2 + (5 - 6)^2)`
= `sqrt((-4)^2 + (-1)^2)`
= `sqrt(16 +1)`
= `sqrt(17)`
BC = `sqrt((2 - 1)^2 + (1 - 5)^2)`
= `sqrt((1)^2 + (-4)^2)`
= `sqrt(1+16)`
= `sqrt(17)`
CD = `sqrt((6 - 2)^2 + (2 - 1)^2)`
= `sqrt((4)^2 + (1)^2)`
= `sqrt(16 + 1)`
= `sqrt(17)`
DA = `sqrt((5 - 6)^2 + (6 - 2)^2)`
= `sqrt((-1)^2 + (4)^2)`
= `sqrt(1+16)`
= `sqrt(17)`
To prove it is a square, the diagonals must also be equal in length.
AC = `sqrt((2 - 5)^2 + (1 - 6)^2)`
= `sqrt((-3)^2 + (-5)^2)`
= `sqrt(9+25)`
= `sqrt(34)`
BD = `sqrt((6 - 1)^2 + (2 - 5)^2)`
= `sqrt((5)^2 + (-3)^2)`
= `sqrt(25 + 9)`
= `sqrt(34)`
The diagonals have the same length, `sqrt(34)` units.
Since, AB = BC = CD = DA and AC = BD,
A, B, C and D are the vertices of a square.
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Tharunya was thrilled to know that the football tournament is fixed with a monthly timeframe from 20th July to 20th August 2023 and for the first time in the FIFA Women’s World Cup’s history, two nations host in 10 venues. Her father felt that the game can be better understood if the position of players is represented as points on a coordinate plane. |
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[or]
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