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

This question was posted earlier, and I had suggested that there might be a typo, since none of the line segments are parallel

http://www.jiskha.com/display.cgi?id=1362226321#1362226321.1362230628

I see it posted exactly the same way, but ....

slope AB = 1/3
slope BC = 1/5
slope CD = 5/1 = 5
slope DA = 5/3

Once you have found your typo and corrected it, follow bobpursley's method.

To find the height of the parallelogram, we need to determine the perpendicular distance between the base AB and the line containing either side CD or AD.

Step 1: Determine the slope of AB.
The slope of AB can be calculated using the formula:
m = (y2 - y1) / (x2 - x1), where (x1, y1) = A(1,-2) and (x2, y2) = B(2,3).
m = (3 - (-2)) / (2 - 1)
m = 5

Step 2: Determine the equation of the line containing either side CD or AD.
We will use side CD for this calculation.

1. Determine the slope of CD.
The slope of CD can be calculated using the formula:
m = (y2 - y1) / (x2 - x1), where (x1, y1) = C(-3,2) and (x2, y2) = D(-4,-3).
m = (-3 - 2) / (-4 - (-3))
m = -5

2. Determine the equation of the line using the slope-intercept form.
y = mx + b, where m = slope and (x, y) = a point on the line.
Using point C(-3, 2):
2 = -5(-3) + b
2 = 15 + b
b = 2 - 15
b = -13

The equation of the line containing side CD is y = -5x - 13.

Step 3: Determine the perpendicular distance between the base AB and the line CD.
The perpendicular distance between a line and a point can be calculated using the formula:
d = |Ax + By + C| / √(A^2 + B^2), where (x, y) = a point on the line and A, B, C are the coefficients of the line equation.

The coefficients of the line CD are A = 5, B = -1, C = -13.

Using the coordinates of point A(1,-2):
d = |(5 * 1) + (-1 * -2) + (-13)| / √(5^2 + (-1)^2)
d = |5 + 2 - 13| / √(25 + 1)
d = |-6| / √26
d = 6 / √26

So, the height of the parallelogram, taking AB as the base, is 6 / √26.

To find the height of a paralellogram, we need to determine the perpendicular distance between the base and the opposite side.

First, let's find the equation of the line passing through points A(1,-2) and B(2,3), which gives us the line segment AB. We can use the formula for the equation of a straight line:

y = mx + b

Where m is the slope of the line and b is the y-intercept.

The slope (m) can be calculated using the formula:

m = (y2 - y1) / (x2 - x1)

Substituting the values of points A and B:

m = (3 - (-2)) / (2 - 1)
m = 5 / 1
m = 5

Now, we can use the point-slope form of the equation to determine the equation of the line:

y - y1 = m(x - x1)

Using point A(1,-2):

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.

Now, let's find the perpendicular distance between the line AB and point C(-3,2).

To find the shortest distance between a point and a line, we can use the formula:

Distance = |Ax + By + C| / sqrt(A^2 + B^2)

Where A, B, and C are the coefficients of the equation of the line, and (x, y) are the coordinates of the point.

Substituting the values:

Distance = |(5)(-3) + (-1)(2) + (-7)| / sqrt((5)^2 + (-1)^2)
Distance = |-15 - 2 - 7| / sqrt(25 + 1)
Distance = 24 / sqrt(26)

Therefore, the height of the parallelogram is given by the perpendicular distance between line AB and point C, which is 24 / sqrt(26) units.

I would take the line DA, write it as y=mx+b, then find the vertical line from it to B. The distance of that is the height.

sketch it so you understand.
a. slope DA=(yd-ya)/(xd-xa)=(-3+2)/(-4-1)= 1/5

Now using that, and the point D, the equation of DA can be found:
y=mx+b
-3=1/5 (-4)+b or b=-3+4/5= you do it.

Now, the line perpendicular to DA going through the point C is what we are going after. Slope of that line is -5.
y=-5x+b but you know point C is in it
so, 2=-5(2)+b so b=12
and then this means the equation of the altitude is y=-5x+12
Now we need the distance of the altitude. Where does it intersect DA?
y=-5x+12 and
y=-1/5 x -3+4/5
solve for x and y, that is the point of intersection.

Now use the distance formula to find the distance between C and the intersection point, that is the altitude.