Three vertices of a quadrilatral are
(-1 -1), (1, 2), and (5, -1). What are the coordinate of two vertices that will form two different parallelograms?
To find two different parallelograms using the given three vertices (-1, -1), (1, 2), and (5, -1), we need to select any two pairs of opposite sides that are parallel to each other. Here's how you can find the coordinates of the two additional vertices:
Step 1: Calculate the differences in x and y coordinates between the two given vertices to find the vector representing one side of the parallelogram.
- For the first pair of vertices, (-1, -1) and (1, 2), the difference is (1 - (-1), 2 - (-1)) = (2, 3).
- For the second pair of vertices, (1, 2) and (5, -1), the difference is (5 - 1, -1 - 2) = (4, -3).
Step 2: Since we want to find two different parallelograms, we can multiply the vector found in step 1 by any non-zero scalar value and add/subtract it to/from the given points to obtain two sets of new coordinates for the other two vertices:
Option 1: Multiply the vector (2, 3) by any non-zero scalar, say k = 2:
- For the first parallelogram, we add the scaled vector to the first given vertex (-1, -1). The coordinates of the new vertex are (-1, -1) + 2(2, 3) = (-1, -1) + (4, 6) = (3, 5).
- For the second parallelogram, we subtract the scaled vector from the second given vertex (1, 2). The coordinates of the new vertex are (1, 2) - 2(2, 3) = (1, 2) - (4, 6) = (-3, -4).
Option 2: Multiply the vector (4, -3) by any non-zero scalar, say k = -3:
- For the third parallelogram, we add the scaled vector to the first given vertex (-1, -1). The coordinates of the new vertex are (-1, -1) + (-3)(4, -3) = (-1, -1) + (-12, 9) = (-13, 8).
- For the fourth parallelogram, we subtract the scaled vector from the second given vertex (1, 2). The coordinates of the new vertex are (1, 2) - (-3)(4, -3) = (1, 2) - (-12, 9) = (13, -7).
Therefore, the coordinates of the two additional vertices that can form different parallelograms with (-1, -1), (1, 2), and (5, -1) are: (3, 5), (-3, -4), (-13, 8), and (13, -7).