The midpoint of P(1,2) and A is Q; the midpoint of P and C is S; the midpoint of Q and S is R(5,1.5). Find the coordinates of B, the midpoint of A and C.
Let's say A has co-ordinates (a,b),
Q has co-ordinates (q,r)
C has co-ordinates (c,d),
S has co-ordinates (s,t)
and B has co-ordinates (x,y).
When any point (m,n) is the midpoint of (f,g) and (h,i), then:
(f+h)/2 = m, and (g+i)/2 = n
So,
i) Q is the midpoint of P and A
=> (a+1)/2 = q , (b+2)/2 = r
ii) S is the midpoint of P and C
=> (c+1)/2 = s , (d+2)/2 = r
iii) R is the midpoint of Q and S
=> ((a+1)/2 + (c+1)/2)/2 = 5
and
((b+2)/2 + (d+2)/2)/2 = 1.5
=> (a+1) + (c+1) = 20
and
(b+2) + (d+2) = 6
=> a + c = 18
and
b + d = 2
We have to find (x,y)
x = (a+c)/2, y = (b+d)/2
=> x = 18/2, y = 2/2
=> x = 9, y = 1