Midpoints Of A Rhombus
Geometry
Level
2
Take a rhombus and connect the midpoints. Which of the following best describes the shape that we get?
Square
Rhombus
Rectangle
Parallelogram
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1 solution
Put the rhombus on a coordinate plane with the left vertex at (0,0), right vertex at ( c, 0), top vertex at (a,b) and bottom vertex at ( a , -b). Using the midpoint theorem, the midpoints are at (a/2,b/2), ((c+a)/2 , b/2), ((c+a)/2, -b/2), and (a/2, -b/2). Since all x values on the left are a/2 and on the right are (c+a)/2, the left and right segments are parallel and vertical. Since the top midpoints have the same y value b/2 and the bottom midpoints have the same y value - b/2, the top and bottom segments are parallel and horizontal. Therefore the figure is a parallelogram and since vertex angles are 90 degrees it is a rectangle. The horizontal length is c/2 and the vertical length is b. We cannot assume they are equal so rectangle.