Prove that the points (2, -1), (0, 2), (2, 3) and (4, 0) are the coordinates of the vertices of a parallelogram and find the angle between its diagonals.

To prove:

The points (2, -1), (0, 2), (2, 3) and (4, 0) are the coordinates of the vertices of a parallelogram

To find:

The angle between diagonals of parallelogram.

Assuming:

A(2, − 1), B(0, 2), C(2, 3) and D(4, 0) be the vertices.

Explanation:

Slope of AB

Slope of BC

Slope of CD

Slope of DA

Thus, AB is parallel to CD and BC is parallel to DA.

Therefore, the given points are the vertices of a parallelogram.

Now, let us find the angle between the diagonals AC and BD.

Let m₁ and m₂ be the slopes of AC and BD, respectively.

∴

∴

Thus, the diagonal AC is parallel to the y-axis.

∴∠ODB

In triangle MND,

∠DMN

Hence proved, the acute angle between the diagonal is .