Note: Your teacher will grade your response to this question to ensure you receive proper credit for your answer.%0D%0AThe coordinates for a rhombus are given as (2a, 0), (0, 2b), (–2a, 0), and (0, –2b). Write a plan to prove that the midpoints of the sides of a rhombus determine a rectangle using coordinate geometry. Be sure to include the formulas.%0D%0A%0D%0AThe upper and lower vertices of the rhombus are located on the y-axis.%0D%0Athe left and right vertices are located on the x-axis.%0D%0AThe upper vertex has coordinates left parentheses 0 comma 2 b right parentheses.%0D%0AThe midpoint of the upper right side has coordinates left parentheses a comma b right parentheses.%0D%0AThe right vertex has coordinates left parentheses 2 a comma 0 right parentheses.%0D%0AThe midpoint of the lower right side has coordinates left parentheses a comma negative b right parentheses.%0D%0AThe lower vertex has coordinates left parentheses 0 comma negative 2 b right parentheses.%0D%0AThe midpoint of the lower left side has coordinates left parentheses negative a comma negative b right parentheses.%0D%0AThe left vertex has coordinates left parentheses negative 2 a comma 0 right parentheses.%0D%0AThe midpoint of the upper left side has coordinates left parentheses negative a comma b right parentheses.%0D%0A(3 points)

To prove that the midpoints of the sides of a rhombus determine a rectangle, we will follow these steps:

1. Find the slope of two opposite sides of the rhombus:
- The slope of the upper side is given by the formula:
m_upper = (y2 - y1) / (x2 - x1), where (x1, y1) = (2a, 0) and (x2, y2) = (0, 2b)
Plugging in the values, we get:
m_upper = (2b - 0) / (0 - 2a) = -b/a

- The slope of the lower side is given by the formula:
m_lower = (y2 - y1) / (x2 - x1), where (x1, y1) = (-2a, 0) and (x2, y2) = (0, -2b)
Plugging in the values, we get:
m_lower = (-2b - 0) / (0 - (-2a)) = -b/a

Note: The slopes are equal, as -b/a = -b/a. This shows that the opposite sides of the rhombus are parallel.

2. Find the slope of another two opposite sides of the rhombus:
- The slope of the right side is given by the formula:
m_right = (y2 - y1) / (x2 - x1), where (x1, y1) = (2a, 0) and (x2, y2) = (0, -2b)
Plugging in the values, we get:
m_right = (-2b - 0) / (0 - 2a) = b/a

- The slope of the left side is given by the formula:
m_left = (y2 - y1) / (x2 - x1), where (x1, y1) = (-2a, 0) and (x2, y2) = (0, 2b)
Plugging in the values, we get:
m_left = (2b - 0) / (0 - (-2a)) = b/a

Note: The slopes are equal, as b/a = b/a. This shows that the other opposite sides of the rhombus are parallel.

3. Calculate the perpendicular slopes of the sides:
- The perpendicular slope of m_upper is m_perp_upper = -1/m_upper
Substituting -b/a for m_upper, we get:
m_perp_upper = -1 / (-b/a) = a/b

- The perpendicular slope of m_lower is m_perp_lower = -1/m_lower
Substituting -b/a for m_lower, we get:
m_perp_lower = -1 / (-b/a) = a/b

- The perpendicular slope of m_right is m_perp_right = -1/m_right
Substituting b/a for m_right, we get:
m_perp_right = -1 / (b/a) = -a/b

- The perpendicular slope of m_left is m_perp_left = -1/m_left
Substituting b/a for m_left, we get:
m_perp_left = -1 / (b/a) = -a/b

Note: The perpendicular slopes are equal, as a/b = a/b. This shows that the sides of the rhombus are perpendicular to each other.

4. Verify if the midpoints of the sides form a rectangle:
- The coordinates of the midpoints are (a, b), (a, -b), (-a, -b), and (-a, b).

- The slope between the midpoints (a, b) and (a, -b) is (b - (-b)) / (a - a) = 2b/0 = undefined.

- The slope between the midpoints (a, -b) and (-a, -b) is (-b - (-b)) / (-a - a) = 0/(-2a) = 0.

- The slope between the midpoints (-a, -b) and (-a, b) is (-b - b) / (-a - (-a)) = -2b/0 = undefined.

- The slope between the midpoints (-a, b) and (a, b) is (b - b) / (a - (-a)) = 0/(2a) = 0.

Note: The slopes between the midpoints are undefined or equal to 0, which implies that the sides are perpendicular to each other. This shows that the midpoints of the sides of the rhombus determine a rectangle.

Therefore, the plan to prove that the midpoints of the sides of a rhombus determine a rectangle using coordinate geometry involves calculating the slopes of the sides and verifying that they are parallel and perpendicular to each other. By examining the slopes between the midpoints, we can determine if the figure formed is a rectangle.