The 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.
To prove that the midpoints of the sides of a rhombus determine a rectangle using coordinate geometry, we can follow the following plan:
Step 1: Find the coordinates of the midpoints of the rhombus.
- Given the coordinates for a rhombus as (2a, 0), (0, 2b), (-2a, 0), and (0, -2b), we can find the coordinates of the midpoints by averaging the x-coordinates and the y-coordinates separately.
Midpoint of the first side: ((2a + 0) / 2, (0 + 2b) / 2) = (a, b)
Midpoint of the second side: ((0 + (-2a)) / 2, (2b + 0) / 2) = (-a, b)
Midpoint of the third side: ((-2a + 0) / 2, (0 + (-2b)) / 2) = (-a, -b)
Midpoint of the fourth side: ((0 + 2a) / 2, (-2b + 0) / 2) = (a, -b)
Step 2: Calculate the slopes of adjacent midpoints.
- To prove that the midpoints form a rectangle, we need to show that the slopes of adjacent midpoints are negative reciprocals.
Slope between the first and second midpoint: (b - b) / (a - (-a)) = 0 / (2a) = 0
Slope between the second and third midpoint: (b - (-b)) / (-a - (-a)) = (2b) / 0 = undefined (because division by zero is undefined)
Slope between the third and fourth midpoint: ((-b) - (-b)) / (a - a) = 0 / 0 = undefined
Slope between the fourth and first midpoint: ((-b) - b) / (a - a) = (-2b) / 0 = undefined
Step 3: Calculate the length and slopes of diagonals.
- To prove that the rhombus is a rectangle, we need to show that the diagonals are congruent.
Length of the first diagonal: √((2a - (-2a))^2 + (0 - 0)^2) = √(4a^2) = 2a
Length of the second diagonal: √((0 - 0)^2 + (2b - (-2b))^2) = √(16b^2) = 4b
Step 4: Determine if the rhombus satisfies the conditions of a rectangle.
- A rectangle has opposite sides that are congruent and perpendicular to each other. We already established that the diagonals are congruent in Step 3. Now we need to check the slopes of adjacent sides.
Slope between the first and third midpoint: (b - (-b)) / (a - (-a)) = (2b) / (2a) = b/a
Slope between the second and fourth midpoint: (b - (-b)) / (-a - a) = (2b) / (-2a) = -b/a
Since the slopes of adjacent sides are negative reciprocals (b/a and -b/a), the midpoints of the sides of the rhombus determine a rectangle.