prove that the points (2,-1) (0,2) (2,3)and (4,0) are the coordinates of vertices of a parallelogram and find the angle between its diagonals
If we label the points ABCD as given, then
AB and CD have slope -3/2
AD and BC have slope 1/2
so, the sides are parallel
AC is vertical
BD has slope -1/2
so, tanθ = 2 ==> θ = 63.4°
The calculations are easier if you plot the points and draw the diagonals.
To prove that the given points form the vertices of a parallelogram, we need to show that both pairs of opposite sides are parallel and equal in length.
Let's start by finding the slopes of the sides.
Slope of AB: (y2 - y1) / (x2 - x1) = (2 - (-1)) / (0 - 2) = 3 / (-2) = -3/2
Slope of BC: (y2 - y1) / (x2 - x1) = (3 - 2) / (2 - 0) = 1 / 2
Slope of CD: (y2 - y1) / (x2 - x1) = (0 - 3) / (4 - 2) = -3 / 2
Slope of DA: (y2 - y1) / (x2 - x1) = (-1 - 0) / (2 - 4) = 1 / (-2) = -1/2
Since the slopes of AB and CD are equal (-3/2), and the slopes of BC and DA are equal (1/2 and -1/2 respectively), we can conclude that the opposite sides are parallel.
Now let's calculate the lengths of the sides.
Length of AB: √((x2 - x1)^2 + (y2 - y1)^2) = √((0 - 2)^2 + (2 - (-1))^2) = √(4 + 9) = √13
Length of BC: √((x2 - x1)^2 + (y2 - y1)^2) = √((2 - 2)^2 + (3 - 2)^2) = √1 = 1
Length of CD: √((x2 - x1)^2 + (y2 - y1)^2) = √((4 - 2)^2 + (0 - 3)^2) = √(4 + 9) = √13
Length of DA: √((x2 - x1)^2 + (y2 - y1)^2) = √((2 - 4)^2 + (-1 - 0)^2) = √5
Since both pairs of opposite sides have equal lengths (AB = CD = √13, BC = DA = 1), we can conclude that the given points form the vertices of a parallelogram.
To find the angle between the diagonals, we can use the fact that the diagonals of a parallelogram bisect each other.
Let's find the midpoint of AC and BD:
Midpoint of AC: ((x1 + x2) / 2, (y1 + y2) / 2) = ((2 + 2) / 2, (-1 + 3) / 2) = (2, 1)
Midpoint of BD: ((x1 + x2) / 2, (y1 + y2) / 2) = ((0 + 4) / 2, (2 + 0) / 2) = (2, 1)
We can see that the midpoints of AC and BD are both (2, 1).
Now, we can find the slopes of AC and BD:
Slope of AC: (1 - (-1)) / (2 - 2) = 2 / 0 = Undefined (Vertical line)
Slope of BD: (1 - 1) / (2 - 2) = 0 / 0 = Undefined (Horizontal line)
Since the slopes of AC and BD are undefined, we can conclude that AC is vertical and BD is horizontal. Therefore, the angle between the diagonals is 90 degrees or a right angle.
In conclusion, the given points (2,-1), (0,2), (2,3), and (4,0) form the vertices of a parallelogram, and the angle between its diagonals is 90 degrees or a right angle.