the point (3,-4,5), (1,0,5), and (3,1,-2) are three out of four vertices of a parallelogram ABCD. Explain why there are three possibilities for the location of the fourth vortex, and find the point?
the three vertices determine a plane. The 4th point must lie in that plane, but you can pick any of the three points to be the "origin" relative to the other two.
To understand why there are three possibilities for the location of the fourth vertex of the parallelogram ABCD, we need to know some properties of parallelograms.
1. Opposite sides of a parallelogram are parallel and have the same length.
2. Diagonals of a parallelogram bisect each other.
For the given information, we have three vertices: A(3,-4,5), B(1,0,5), and C(3,1,-2). Let's call the missing vertex D(x, y, z).
First, we can determine one possible location for the fourth vertex by using the property that opposite sides of a parallelogram are parallel and have the same length. We can take AB as one side of the parallelogram and then extend that same vector from point C. So, we have AB = CD.
Let's calculate AB: AB = B - A = (1, 0, 5) - (3, -4, 5) = (-2, 4, 0).
To find the coordinates of point D, we add the vector AB to point C: D = C + AB = (3, 1, -2) + (-2, 4, 0) = (1, 5, -2).
So, one possibility for the location of the fourth vertex is D(1, 5, -2).
To understand the other two possibilities, we need to use the property that the diagonals of a parallelogram bisect each other. Let's consider the diagonals AC and BD.
The midpoint of AC is M, and it is given by the average of the coordinates of A and C: M = ( (3+3)/2, (-4+1)/2, (5-2)/2 ) = (3, -3/2, 3/2).
Since opposite sides of a parallelogram are parallel and have the same length, and AC is a diagonal, then the midpoint of BD, N, must also be (3, -3/2, 3/2).
Using the midpoint N and one of the given vertices as a reference, we can calculate the vector going from N to that vertex. Let's use the vertex B.
NB = B - N = (1, 0, 5) - (3, -3/2, 3/2) = (-2, 3/2, 5/2).
Now, add vector NB to the missing vertex C to find the other two possibilities for the fourth vertex:
D1 = C + NB = (3, 1, -2) + (-2, 3/2, 5/2) = (1, 5/2, 1/2).
D2 = C + NB = (3, 1, -2) + (-2, 3/2, 5/2) = (1, 5/2, 1/2).
Therefore, the other two possibilities for the location of the fourth vertex are D1(1, 5/2, 1/2) and D2(1, 5/2, 1/2).