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.
A rhombus is drawn on a coordinate plane and the midpoints of all 4 sides are plotted.
The coordinates for the four vertices of the rhombus are:
A = (2a, 0)
B = (0, 2b)
C = (-2a, 0)
D = (0, -2b)
The midpoint of AB is:
MAB = ((2a + 0)/2, (0 + 2b)/2) = (a, b)
The midpoint of BC is:
MBC = ((0 - 2a)/2, (2b + 0)/2) = (-a, b)
The midpoint of CD is:
MCD = ((-2a + 0)/2, (0 - 2b)/2) = (-a, -b)
The midpoint of DA is:
MDA = ((0 + 2a)/2, (-2b + 0)/2) = (a, -b)
We can now use the distance formula to show that the sides of the rhombus have equal length. For example:
AB = sqrt((2a - 0)^2 + (0 - 2b)^2) = sqrt(4a^2 + 4b^2)
BC = sqrt((0 + 2a)^2 + (2b - 0)^2) = sqrt(4a^2 + 4b^2)
CD = sqrt((-2a - 0)^2 + (0 + 2b)^2) = sqrt(4a^2 + 4b^2)
DA = sqrt((0 - 2a)^2 + (-2b - 0)^2) = sqrt(4a^2 + 4b^2)
Therefore, the rhombus is equilateral.
Now we need to show that the diagonals of the rhombus are perpendicular. The diagonals of a rhombus are the line segments that connect opposite vertices. The diagonals of this rhombus are AC and BD.
The coordinates of the midpoint of AC are:
MAC = ((2a - 2a)/2, (0 + 0)/2) = (0, 0)
The coordinates of the midpoint of BD are:
MBD = ((0 - 0)/2, (2b - (-2b))/2) = (0, 0)
Since both midpoints are at the origin, the diagonals AC and BD intersect at the origin. Also, since the diagonals of a rhombus bisect each other at right angles, we need to show that AC and BD have slopes that are negative reciprocals of each other.
The slope of AC is:
mAC = (0 - 0)/(2a - (-2a)) = 0
The slope of BD is:
mBD = (2b - (-2b))/(0 - 0) = undefined
Since the slope of AC is 0 and the slope of BD is undefined, we can see that AC is a horizontal line and BD is a vertical line. Therefore, AC and BD meet at a right angle, which means that the midpoints of the sides of the rhombus determine a rectangle.
To summarize:
1. Show that the sides of the rhombus have equal length using the distance formula.
2. Show that the diagonals of the rhombus are perpendicular by showing that they meet at a right angle and have slopes that are negative reciprocals of each other.
3. Conclude that the midpoints of the sides of the rhombus determine a rectangle.