What type of figure is formed by joining the midpoints of the sides of a parallelogram? Give a careful explanation of your answer.

Answer 1

Hi Cass! Great question. This might be hard to explain without a drawing, but I'll give it a shot.

Imagine we have a parallelogram with sides A, B, C, D, (in order going clockwise) and let's refer to the midpoints of those sides as a, b, c, and d. Because our shape is a parallelogram, we know by definition that sides A and C are parallel, and sides B and D are parallel. What we'll see is that connection our midpoints gives us two sets of parallel lines!

When we connect points (a,b) and (d,c), we can see they're traversing the same rise-over-run distance (half of side A, half of side C respectively) because A and C are the same length. Similarly, when we connect (b,c) and (a,d), we'll get two more parallel lines. Since our shape has two sets of parallel lines, it must be a parallelogram!

Depending on your knowledge of geometry, we can dive deeper into this answer. We can in fact prove that our pairs of lines, say (a,b) and (d,c), are parallel using similar triangles.