We start with square WXYZ. To make the new quadrilateral, connect the midpoints of WX, XY, YZ and ZW in that order. We'll show the shape you get is a square.
Easy coordinate proof (good for a 13-year-old):
- Put the square on a coordinate grid so its vertices are easy numbers. Let the side length be s and choose W = (0, 0), X = (s, 0), Y = (s, s), Z = (0, s).
- Find the midpoints of the sides and name them P, Q, R, S in order:
- P, midpoint of WX = ((0 + s)/2, (0 + 0)/2) = (s/2, 0)
- Q, midpoint of XY = ((s + s)/2, (0 + s)/2) = (s, s/2)
- R, midpoint of YZ = ((s + 0)/2, (s + s)/2) = (s/2, s)
- S, midpoint of ZW = ((0 + 0)/2, (s + 0)/2) = (0, s/2)
- Compute one side length, for example PQ: PQ = distance between (s/2, 0) and (s, s/2) = sqrt((s - s/2)^2 + (s/2 - 0)^2) = sqrt((s/2)^2 + (s/2)^2) = sqrt(s^2/4 + s^2/4) = sqrt(s^2/2) = s/√2. By symmetry the other three sides QR, RS, SP have the same length.
- Check a right angle, for example at Q using slopes: slope of PQ = (s/2 - 0)/(s - s/2) = (s/2)/(s/2) = 1, slope of QR = (s - s/2)/(s/2 - s) = (s/2)/(-s/2) = -1. Slopes 1 and -1 multiply to -1, so PQ is perpendicular to QR. So adjacent sides are perpendicular.
- Having four equal sides and a right angle between adjacent sides means PQRS is a square.
A short conceptual explanation (Varignon's theorem):
Varignon's theorem says that joining the midpoints of the sides of any quadrilateral produces a parallelogram. In the special case where the original quadrilateral is a square, the parallelogram you get has equal sides (a rhombus) and its adjacent sides are perpendicular, so the parallelogram is actually a square.
Conclusion: The quadrilateral formed by joining the midpoints of the sides of square WXYZ is itself a square (rotated 45° relative to the original and with side length s/√2 if the original had side s).