Question

HELP QUICK!!!!
REASONING Points (1, 2), (3, 6), and (6, 4) are three vertices of a parallelogram. How many parallelograms can be created using these three vertices? Find the coordinates of each point that could be the fourth vertex.

Answer 1

Answer:

3 parallelograms

Fourth vertex: (-2, 4) or (4, 0) or (8, 8)

Step-by-step explanation:

To create parallelograms using the three given vertices, we need to find the coordinates of the fourth vertices such that the opposite sides are parallel and equal in length.

Let the three vertices be:

• A(1, 2)
• B(3, 6)
• C(6, 4)

If we connect points B and C with a line segment, and draw a line parallel to this that passes through point A, we can create two parallelograms where the fourth vertices lie on this line:

• The fourth vertex of the first parallelogram (D₁) is the point of intersection of this line and the line parallel to line segment AC that passes through point B (see attachment 1). The coordinates of this vertex are (-2, 4).

• The fourth vertex of the second parallelogram (D₂) is the point of intersection of this line and the line parallel to line segment AB that passes through point C (see attachment 2). The coordinates of this vertex are (4, 0).

Similarly, if we connect points A and C with a line segment, and draw a line parallel to this that passes through point B, we can create a third parallelogram where the fourth vertex lies on this line:

• The fourth vertex of the third parallelogram (D₃) is the point of intersection of this line and the line parallel to line segment AB that passes through point C (see attachment 3). The coordinates of this vertex are (8, 8).