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A(-1,1)B(5,3) are opposite vertices of a square. Perpendicular distance from (1,2) to other diagonal (wh Perpendicular ich is not passing through A,B) of the square

A(-1,1)B(5,3) are opposite vertices of a square. Perpendicular distance from (1,2) to other diagonal (wh Perpendicular ich is not passing through A,B) of the square

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Grade:11

1 Answers

Samyak Jain
333 Points
5 years ago
A(-1,1) , B(5,3) are opposite vertices of a square, say ACBD.
Slope of diagonal AB = (3 – 1)/(5 + 1) = 1/3
Let m1 = 1/3 and m2 = slope of diagonal CD.
\because diagonals of a square are perpendicular to each other, \therefore m1.m2 = – 1
i.e. 1/3 . m2 = – 1  \Rightarrow m2 = – 3
Coordinates of mid-point of AB are (( – 1+ 5)/2 , (1 +3 )/2)
i.e. (2,2).
Diagonals of a square bisect each other. 
So (2,2) lies on CD also.
\therefore Equation of CD is y – 2 = (–3)(x-2)
\Rightarrow 3x + y – 8 = 0.
We know that perpendicular distance from a point (x1,y1) to a line ax+b+c=0
is |(ax1 + by1 + c)/\sqrt{a^{2} + b^{2}}|.
Thus, perpendicular distance from point (1,2) to  line 3x + y – 8 = 0
is |(3 + 2 – 8)/\sqrt{3^{2} + 1^{2}}|  =  |5 – 8| / \sqrt{10}  
   =  3 / \sqrt{10}  

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