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प्रश्न
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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उत्तर
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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Use of mobile screen for long hours makes your eye sight weak and give you headaches. Children who are addicted to play "PUBG" can get easily stressed out. To raise social awareness about ill effects of playing PUBG, a school decided to start 'BAN PUBG' campaign, in which students are asked to prepare campaign board in the shape of a rectangle: One such campaign board made by class X student of the school is shown in the figure.
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