Solution:
Vertices of parallelogram ABCD is given as A(1,2) , B(0,9) , C(7,8) , and D(8,1) .
We will use the following analytical geometry formulas here
1. Distance between two points

2. Slope of a line

3. When two lines are perpendicular, product of their slopes is equal to -1.
Since ABCD is a parallelogram
1. Opposite sides are equal and parallel.
2. Diagonals bisect each other.
3. Opposite angles are equal.
Now, coming to problem

As, you can see that, AB=BC=CD=DA=√ 50
But , slope of AB × Slope of BC =slope of CB × Slope of DC=slope of CD × Slope of DA
![=[-7 * (-1)/(7)]=1](https://img.qammunity.org/2020/formulas/mathematics/college/b24o6yjzmhzhpjjz5ptetsrnqrwdn09lgo.png)
which is not equal to -1. It means lines which are sides of parallelogram are not perpendicular.
As, all side of parallelogram ABCD are equal, so it is a rhombus.
As, diagonal of rhombus bisect each other at right angles.

Shows that diagonals are perpendicular bisector of each other.
Option (1) : AC⊥BD; therefore, ABCD is a rhombus.