Question

Four members of a marching band notice that they've formed the vertices of a parallelogram, if the field they were on had coordinates like the coordinate plane. Cindy is at (-2, 2), Arturo is at (3, 1), and Rowan is at (-1, 4).

Part 1: What are all possible coordinate locations for Hailey, the fourth band member?

Part 2: How do you know that you have found all possible coordinates? Justify your answer using words, mathematics, and/or patterns as needed.

Answer 1

Part 1:
To find all possible coordinate locations for Hailey, we need to use the fact that the vertices of a parallelogram have opposite sides that are parallel and equal in length. We can use vector addition and subtraction to find the missing coordinate for Hailey.

Let AB be the vector from Arturo to Cindy, and let AR be the vector from Arturo to Rowan. Then, the vector from Cindy to Rowan is BR = AR - AB = (-1-3, 4-1) = (-4, 3).

Since the opposite sides of a parallelogram are parallel and equal in length, the vector from Hailey to Rowan must be BR, and the vector from Hailey to Cindy must be AB.

Let HR be the vector from Hailey to Rowan, and let HC be the vector from Hailey to Cindy. Then, we have the following system of equations:

HR = (-4, 3)
HC = (-2-x, 2-y)
HR + AB = HC
HR + AR = AB

Substituting the values for AB and AR, we get:

HR + (5, -1) = (-2-x, 2-y)
HR + (-4, 3) = (5, -1)

Solving for HR, we get:

HR = (-1-x, -1+y)

Substituting into the first equation and solving for HC, we get:

HC = (-3-x, 1+y)

Therefore, all possible coordinate locations for Hailey are of the form (x, y), where x and y are integers such that (-3-x, 1+y) is not equal to (-2, 2) and (-1-x, -1+y) is equal to (-4, 3).

Part 2:
We know that we have found all possible coordinates for Hailey because we have used the fact that the vertices of a parallelogram have opposite sides that are parallel and equal in length. This means that once we find one pair of opposite sides, we can use vector addition and subtraction to find the missing coordinates. Since there are only two sides, we can be sure that we have found all possible coordinates by checking that the coordinates we found satisfy both equations.