To find the value of a and the height of the parallelogram, we can use the property that the opposite sides of a parallelogram are parallel and equal in length.

1. Finding the length of side AB:
Using the distance formula, we can find the length of side AB:
AB = √[(x2 - x1)^2 + (y2 - y1)^2]
AB = √[(2 - 1)^2 + (3 - (-2))^2]
AB = √[1^2 + 5^2]
AB = √[1 + 25]
AB = √26

2. Finding the equation of line AB:
The equation of a line passing through two points (x1, y1) and (x2, y2) is given by:
(y - y1) = m(x - x1)
Using the points A(1, -2) and B(2, 3), we can find the slope of line AB:
m = (y2 - y1)/(x2 - x1)
m = (3 - (-2))/(2 - 1)
m = (3 + 2)/(2 - 1)
m = 5/1
m = 5
Substituting the slope and the coordinates of point A into the equation, we get:
(y - (-2)) = 5(x - 1)
y + 2 = 5x - 5
y = 5x - 5 - 2
y = 5x - 7
So, the equation of line AB is y = 5x - 7.

3. Using the property of parallel lines:
Since opposite sides of a parallelogram are parallel, the slope of line CD should also be 5.
The equation of line CD can be written as:
(y - 2) = 5(x - a)
y - 2 = 5x - 5a
y = 5x - 5a + 2

4. Setting the slopes equal to each other:
Since the slopes of AB and CD are equal, we can equate the coefficients of x and solve for a:
5 = 5
Using the coefficient of x, we get:
-7 = -5a + 2
-7 - 2 = -5a
-9 = -5a
a = -9/-5
a = 9/5
So, the value of a is 9/5.

5. Finding the height of the parallelogram:
The height of the parallelogram is the perpendicular distance between the base AB and the line CD. We can find this distance by finding the perpendicular distance between the line AB and the point C.
Using the formula for the distance between a point (x1, y1) and a line Ax + By + C = 0:
d = |Ax1 + By1 + C| / √(A^2 + B^2)
In this case, the equation of line AB is y = 5x - 7, which can be rewritten as 5x - y - 7 = 0.
Substituting the coordinates of point C (a, 2) into the formula, we get:
d = |5a - 2 - 7| / √(5^2 + (-

A = -3