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 
.