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Question

If two opposite vertices of a square are (1, 2) and (5, 8), find the coordinates of its other two vertices and the equations of its sides.


Solution

Let A(1, 2), C(5, 8), B(x1,y1), D(x2,y2)

Slope of AC =8251=64=32

Let m be the slope of a line making an angle 45 with AC.

tan 45=m1321+m×32

1=m321+3m2

1+3m2=m32 or, 1+3m2=(m32)

3m2m=321 or, 1+3m2=m+32

12m=52 or, 3m2+m=321

m=5 or, 5m2=12

m=15

Consider the following figure:

Hence, equation of AD.

y2=5(x1)

y2=5x+5

5x+y=7

Equation of CD,

y8=15(x5)

y8=x51y8=x55

=5y40=x55yx=35

x5y+35=0

Hence co-ordinates are (6, 3), (0, 7)

The equation of AB is

y2=15(x1)

5y10=x1

x5y+9=0

And the equation of BC is

y8=5(x5)

y8=5x+25

5x+y33=0


Mathematics
RD Sharma
Standard XI

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