AS level maths help!

[tanmaydube]

I cannot solve this question can anybody help me in this chapter- coordinates, points and lines
this question is from the book pure mathematics 1, chapter 1, miscellaneous exercise 1, question no. 4, part (b)

A(7,2) and C(1,4) are the two vertices of a square ABCD.
(a) Find the equation of the diagonal BD.
(b) Find the coordinates of B and of D.

thank you for helping.

 

[mathgenius]

the diagonals of square bisect each other at right angles.. so the midpoint of AC is also the mid point of BD
so. midpoint of AC is:
x=(7+ 1)/2 =4 y=(2+4)/2 =3 so the coordinate of midpoint is (4,3)
now the gradient of AC =(Y2-Y1)/(X2-X1) = (4-2)/(1-7) = -1/3
since AC and BD are perpendicular to each other so the gradient of BD is 3
now equation of BD is : (Y-Y1)=m(X-X1)
or,(Y-3)=3(X-4)
or, y-3=3x-12
so the equation is y=3x-9

 

[mathgenius]

now similarly find the equation of AB and find the intersection of AB and BD which gives you the coordinate of B...
and now find the coordinate of D by using the midpoint formula......
good luck