Let A(1, 2), B(4, y), C(x, 6) and D(3, 5) are the vertices of a parallelogram.
We know that the diagonals of parallelogram bisect each other. So,
Coordinates of the mid-point of AC
= Coordinates of the mid-point of BD


Hence, required values are x = 6 and y = 3.

Let the given points he A(1, -5) and B(-4, 5)
Let the x-axis cuts AB at the point P in the ratio K : 1.
Then the co-ordinates of P are given as

Here, we have
x₁ = 1, y₁ = -5
x₂ = -4, y₂ = 5
and    m₁ = K, m₂ = 1
So, coordinates of P are


Since, P lies on x-axis. So its ordinate is 0.

Hence, required ratio = (1 : 1).
Now, the coordinates of the point of division be

Let P(-1, 6) divides the line segment joining the points A(-3, 10) and B(6, -8) in the ratio K : 1.
Then the co-ordinates of P are given by

Here, we have
x₁ = -3, y₁ = 10
x₂ = 6, y₂ = -8
and    m₁ = K, m₂ = 1
So, the co-ordinates of P are



But the co-ordinates of P are given as P(-1, 6)


Hence, required ratio = 2 : 7.

Take A as origin, AB as x-axis and AD as y-axis. Then,
Co-ordinates of the green flag post be



and co-ordinates of the red flag post be

Now, Distance between both the falgs



and Position of the blue flag post be



So, she should post her blue flag on the 5th line at a distance of 22.5 km from AB.

Let P and Q be the point of trisection of AB. Then,

AP = PQ = QB = 1
Case I. Here P divides AB in the ratio 1 : 2.
So, we have
x₁ = 4, y¹ = -1
x_{2 = -2, y² = -3}
and    m₁ = 1, m₂ = 2
∴ The coordinates of P are given by

Case II. Here, Q divides AB in the ratio 2 : 1. So, we have
x₁ = 4, y₁ = -1
x₂ = 2, y₂ = -3
and    m₁ = 2, m₂ = 1
∴ The coordinates of Q are given by