Question

If the points A(6, 1), B(p, 2), C(9, 4) and D(7, 9) are the vertices of a parallelogram ABCD, then find the values of p and g. Hence, check whether ABCD is a rectangle or not.

Answer 1

Given: A(6, 1), B(p, 2), C(9, 4) and D(7, 9)

Midpoint of diagonal AC:

M₁ = `((6+9)/2,(1+4)/2)`

= `(15/2,5/2)`

Midpoint of diagonal BD:

M2 = `((p+7)/2,(2+9)/2)`

= `((p+7)/2,11/2)`

Since M₁ = M₂, we equate the coordinates:

For x-coordinates

`15/2=(p+7)/2`

15 = (p + 7)     ...(Multiplying both sides by 2)

p = 8

For y-coordinates:

`5/2=11/2`

p = 8

A parallelogram is a rectangle if its diagonals are equal in length.

d = `sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

Length of AC: 

AC = `sqrt((9-6)^2+(4-1)^2)`

= `sqrt(3^2+3^2)`

= `sqrt(9+9)`

= `sqrt18`

BD = `sqrt((7-8)^2+(9-2)^2)` 

= `sqrt((-1)^2+7^2)`

= `sqrt(1+49)`

= `sqrt50`

Since AC ≠ BD, the diagonals are not equal.

The value of p is 8.

Since the diagonals are not equal, ABCD is not a rectangle.