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Question
If the point P(x, y ) is equidistant from the points A(5, 1) and B (1, 5), prove that x = y.
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Solution
The distance d between two points `(x_1,y_1)` ``nd `(x_2,y_2)` is given by the formula
`d = sqrt((x_1 - x_2)^2 + (y_1 - y_2)^2)`
The three given points are P(x, y), A(5,1) and B(1,5).
Now let us find the distance between ‘P’ and ‘A’.
`PA = sqrt((x - 5)^2 + (y - 1)^2)`
Now, let us find the distance between ‘P’ and ‘B’.
`PB = sqrt((x - 1)^2 + (y - 5)^2)`
It is given that both these distances are equal. So, let us equate both the above equations,
PA = PB
`sqrt((x - 5)^2 + (y -1)^2) = sqrt((x - 1)^2 + (y - 5)^2)`
Squaring on both sides of the equation we get,
`(x - 5)^2 + (y - 1)^2 = (x - 1)^2 + (y - 5)^2`
`=> x^2 + 25 - 10x + y^2 + 1 - 2y = x^2 + 1 - 2x + y^2 + 25 - 10y`
`=> 26 - 10x - 2y = 26 - 10y - 2x`
`=> 10y - 2y = 10x - 2x`
`=> 8y = 8x`
=> y = x
Hence we have proved that x = y.
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By distance formula,
PQ = `sqrt(square + (y_2 - y_1)^2`
= `sqrt(square + square)`
= `sqrt(square + square)`
= `sqrt(square + square)`
= `sqrt(125)`
= `5sqrt(5)`
Read the following passage:
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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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Based on the above information, answer the following questions:
- Find the coordinates of the point of intersection of diagonals AC and BD.
- Find the length of the diagonal AC.
-
- Find the area of the campaign Board ABCD.
OR - Find the ratio of the length of side AB to the length of the diagonal AC.
- Find the area of the campaign Board ABCD.
