Question

ABCD is a parallelogram. M is the mid-point of AB and P is a point on diagonal BD such that BP = `1/4` BD. MP produced meets BC at N. Prove that:

1. N is a mid-point of BC.
2. MN = `1/2` AC

Answer 1

Given:

• ABCD is a parallelogram.
• M is the midpoint of AB, i.e, AM = MB.
• P is a point on diagonal BD such that `BP = 1/4 BD`, meaning P divides BD in the ratio 1 : 3.
• The line segment MP is extended to meet BC at N.

We need to prove:

1. N is a midpoint of BC,
2. `MN = 1/2 AC`.

Step 1: Set up a coordinate system

Since ABCD is a parallelogram, let’s place it in the coordinate plane for simplicity.

We assign the following coordinates:

• A(0, 0),
• B(b, 0), 
• D(0, d), 
• C(b, d).

So, the coordinates of the vertices of the parallelogram are:

• A(0, 0),
• B(b, 0), 
• C(b, d), 
• D(0, d).

Step 2: Coordinates of M, P and N

1. Midpoint M:

Since M is the midpoint of AB, the coordinates of M are the average of the coordinates of A(0, 0) and B(b, 0):

`M = ((0 + b)/2, (0 + 0)/2) = (b/2, 0)`

2. Point P:

Point P divides the diagonal BD in the ratio 1 : 3. The coordinates of B are (b, 0) and the coordinates of D are (0, d).

Using the section formula, the coordinates of P are:

`P = ((3b + 0)/4, (3 * 0 + d)/4) = ((3b)/4, d/4)`

3. Equation of line MP:

Now, we will find the equation of line MP.

The slope of MP is given by:

Slope of MP = `(d/4 - 0)/((3b)/4 - b/2)`

= `(d/4)/(b/4)`

= `d/b`

The equation of the line MP in point-slope form is:

`y - 0 = d/b (x - b/2)`

Which simplifies to:

`y = d/b (x - b/2)`

4. Point N on line BC:

The line BC has coordinates B(b, 0) and C(b, d), so it is a vertical line with equation x = b.

Substituting x = b into the equation of line MP, we get:

`y = d/b (b - b/2)`

= `d/b xx b/2`

= `d/2`

Therefore, the coordinates of point N are `(b, d/2)`.

Step 3: Prove that N is a midpoint of BC

The coordinates of B are (b, 0) and the coordinates of C are (b, d).

The midpoint of BC has coordinates:

Midpoint of BC = `(b, (0 + d)/2) = (b, d/2)`

This is exactly the coordinates of N.

Hence, N is a midpoint of BC.

Step 4: Prove that `MN = 1/2 AC`

Now, we will prove that `MN = 1/2 AC`

1. Length of AC: The length of AC is the distance between A(0, 0) and C(b, d).

Using the distance formula:

`AC = sqrt((b - 0)^2 + (d - 0)^2`

= `sqrt(b^2 + d^2)`

2. Length of MN: The length of MN is the distance between `M(b/2, 0)` and `N(b, d/2)`.

Using the distance formula:

`MN = sqrt((b - b/2)^2 + (d/2 - 0)^2`

= `sqrt((b/2)^2 + (d/2)^2`

= `1/2 sqrt(b^2 + d^2)`

Thus, we have:

`MN = 1/2 AC`