Checkpoint: midpoints, symmetry and tiles

A triangle's three medians meet at G. What does G do to each median?

• Halves it at G
• Two parts by the corner, one by the side
• One part by the corner and two by the side

The four side midpoints of a quadrilateral are joined in order. One corner of the original points inwards. What is the new shape?

• Still a parallelogram, since each side pair is half a diagonal
• Not a parallelogram, since the argument needed a tidy shape
• A parallelogram only if the original was one, since like makes like

A slanted parallelogram has no fold lines. Which fact lets a half-turn land it on itself?

• Its diagonals are equal, so a half-turn swaps opposite corners
• Its diagonals bisect each other, so a half-turn swaps opposite corners
• Its diagonals cross at right angles, so a half-turn swaps opposite corners

Why does any quadrilateral tile a floor? Does a corner pushed inwards change that?

• Because the pushed-in corner cancels one of the other three corners
• Because four equal sides can always be arranged
• Because four copies give one corner each, totalling 360°