Two verticies of paralellogram ABCD are A(-6,-1) and B(-5,3). The intersection of the diagonals, E, are (-2, 1/2). Determine the coordinates of D.
The diagonals bisect each other so (-2, 1/2) is the midpoint of both.
So use the midpoint theorem
let (a,b) be the point opposing A(-6,-1)
then (-6+a)/2 = -2
-6+a = -4
a = 2
and
(-1+b)/2 = 1/2
-1+b = 1
b = 2
So the point opposite A is (2,2)
Find the other point in the same way
The diagonals bisect each other, so E is the midpoint of BD.
So, D is just as far from E as E is from B
E-B = (4,3/2)
So, D = E + (4,3/2) = (-2,1/2)+(4,3/2) = (2,2)
Oops. I also found C. I said BE, but actually used AE.
But you can find D in the same way.
To find the coordinates of point D, we can use the property that the diagonals of a parallelogram bisect each other. Since the diagonals intersect at point E, we know that the line segment AE is congruent to the line segment CE, and the line segment BE is congruent to the line segment DE.
First, let's find the midpoint of the line segment AE. The midpoint formula is given by:
Midpoint of a line segment (x, y) = ((x1 + x2)/2, (y1 + y2)/2)
Where (x1, y1) and (x2, y2) are the coordinates of the endpoints.
Using the coordinates A(-6, -1) and E(-2, 1/2), we can find the midpoint of AE:
Midpoint of AE: ((-6 + -2)/2, (-1 + 1/2)/2) = (-8/2, -1/2) = (-4, -1/2)
Next, let's find the midpoint of the line segment BE. Using the coordinates B(-5, 3) and E(-2, 1/2), we can find the midpoint of BE:
Midpoint of BE: ((-5 + -2)/2, (3 + 1/2)/2) = (-7/2, 7/4) = (-3.5, 1.75)
Now that we have the midpoints of AE and BE, we can find the coordinates of D by finding the other endpoints of the line segments DE and CE.
The coordinates of D will be the reflection of the midpoints of AE and BE across the point E.
Let's find the reflection of the midpoint of AE:
Reflecting (-4, -1/2) across E(-2, 1/2) can be done by using the formula:
Reflected point (x, y) = (2 * x_reflection - x_original, 2 * y_reflection - y_original)
Applying the formula, we get:
Reflected point for (-4, -1/2) = (2 * -4 - (-2), 2 * (-1/2) - (1/2)) = (-8 + 2, -1 - 1/2) = (-6, -3/2)
Now, let's find the reflection of the midpoint of BE:
Reflected point for (-3.5, 1.75) = (2 * -3.5 - (-2), 2 * 1.75 - (1/2)) = (-7 + 2, 3.5 - 1/2) = (-5, 3)
Therefore, the coordinates of D are (-6, -3/2).