The coordinates for a rhombus are given as (2a, 0), (0, 2b), (-2a, 0), and (0, -2b).

Question

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.

Answer 1

To prove that the midpoints of the sides of a rhombus determine a rectangle using coordinate geometry, we can follow these steps:

1. Calculate the midpoints of the sides of the rhombus:
- The midpoint of (2a, 0) and (0, 2b) is ((2a + 0)/2, (0 + 2b)/2) = (a, b).
- The midpoint of (0, 2b) and (-2a, 0) is ((0 - 2a)/2, (2b + 0)/2) = (-a, b).
- The midpoint of (-2a, 0) and (0, -2b) is ((-2a + 0)/2, (0 - 2b)/2) = (-a, -b).
- The midpoint of (0, -2b) and (2a, 0) is ((0 + 2a)/2, (-2b + 0)/2) = (a, -b).

2. Calculate the slopes of the opposite sides of the rhombus:
- The slope of the side connecting (2a, 0) and (0, 2b) is (2b - 0)/(0 - 2a) = -b/a.
- The slope of the side connecting (0, 2b) and (-2a, 0) is (0 - 2b)/(-2a - 0) = b/a.
- The slope of the side connecting (-2a, 0) and (0, -2b) is (-2b - 0)/(0 - (-2a)) = -b/a.
- The slope of the side connecting (0, -2b) and (2a, 0) is (0 - (-2b))/(2a - 0) = b/a.

3. Since the product of the slopes of the opposite sides of a rhombus is -1 (slopes are negative reciprocals), we can conclude that the opposite sides are perpendicular to each other.

4. Since the midpoints of the sides of a rhombus form a rectangle if the diagonals bisect each other at right angles, we have already established that the opposite sides are perpendicular, and therefore the midpoints of the sides of a rhombus determine a rectangle.

Therefore, by calculating the midpoints and slopes of the sides, we can prove that the midpoints of the sides of a rhombus determine a rectangle using coordinate geometry.