Lesson Check (6.G.3) 1. The coordinates of points \( M, N \), and \( P \) are \( M(-2,3), N(4,3) \), and \( P(5,-1) \). What coordinates for point \( Q \) make MNPQ a parallelogram?

One way to solve the problem is to use the fact that in a parallelogram, the diagonals bisect each other. This means that the midpoint of diagonal MP is the same as the midpoint of diagonal NQ. Step 1. Find the midpoint of MP.   M = (–2, 3) and P = (5, –1)   Midpoint = ((–2 + 5)/2, (3 + (–1))/2) = (3/2, 2/2) = (1.5, 1) Step 2. Let Q = (x, y). The midpoint of NQ (with N = (4, 3)) must equal (1.5, 1).   Midpoint of NQ = ((4 + x)/2, (3 + y)/2) Set this equal to (1.5, 1):   (4 + x)/2 = 1.5  →  4 + x = 3  →  x = –1   (3 + y)/2 = 1  →  3 + y = 2  →  y = –1 So, Q = (–1, –1). Alternatively, you could use the property that in a parallelogram, the vectors for opposite sides are equal. For example:   vector NP = P – N = (5 – 4, –1 – 3) = (1, –4)   Thus, vector MQ must also equal (1, –4). Since M = (–2, 3), adding the vector (1, –4) gives:   Q = M + (1, –4) = (–2 + 1, 3 – 4) = (–1, –1). Either method shows that the coordinates of point Q that make MNPQ a parallelogram are (–1, –1).