if the points A(-1,2) , B(2,3) , C(-3,2) and D(-4,-3) are the vertices of parallelogram ABCD, then taking AB as the base , find the height of this parallelogram .

I just plotted your points, and following the usual notation of joining

A--->B--->C--->D--->A
I get a convex quadrilateral and I see no parallel lines

check your typing of the coordinates

To find the height of the parallelogram ABCD with base AB, we need to calculate the perpendicular distance between the line containing AB and the vertex C or D. Here's how you can do it:

Step 1: Find the equation of the line containing AB.
- Use the two-point form of a straight line: (y - y1) = (x - x1) * (y2 - y1) / (x2 - x1)

Let's take the coordinates of points A(-1,2) and B(2,3).
Using the two-point form, we get:
(y - 2) = (x - (-1)) * (3 - 2) / (2 - (-1))
Simplifying, we have:
y - 2 = (x + 1) * (1/3) or 3y = x + 3.

Therefore, the equation of the line containing AB is: y = (1/3)x + 1.

Step 2: Find the coordinates of point C or D on the line containing AB.
Let's take C(-3,2) or D(-4,-3). Let's use point D(-4,-3) as an example.
We substitute the x and y coordinates of D into the equation of the line to find the point of intersection.

Substituting x = -4 in y = (1/3)x + 1:
y = (1/3)(-4) + 1
y = -4/3 + 1
y = -4/3 + 3/3
y = -1/3

Therefore, the coordinates of point D(-4,-3) on the line containing AB are (-4, -1/3).

Step 3: Calculate the perpendicular distance between line AB and point D.
- We can use the formula for the distance between a point and a line: |Ax + By + C| / sqrt(A^2 + B^2).

The equation of the line containing AB is y = (1/3)x + 1. In the form Ax + By + C = 0, this can be rewritten as x - 3y + 3 = 0.
Now we'll find the perpendicular distance from point D(-4,-3) to this line:
Distance = |(-4) - 3(-1/3) + 3| / sqrt(1^2 + (-3)^2)
Distance = |-4 + 1 + 3| / sqrt(1 + 9)
Distance = |-4 + 4| / sqrt(10)
Distance = 0

The perpendicular distance between line AB and point D is 0.

Therefore, the height of the parallelogram with base AB is 0.