Q1.
Since ΔABD and ΔACD have equal area, you may wonder — Can we divide ΔABD using straight cuts into two or more pieces that we can then rearrange to exactly cover ΔACD? What do you think? Is it possible?
Answer
Yes, it is possible — and here just one straight cut suffices.
AD is a median, so BD = DC and the two triangles have
equal bases and the same height h from A.
ar(ΔABD) = ½ · BD · h = ½ · DC · h = ar(ΔACD)
equal bases and the same height h from A.
ar(ΔABD) = ½ · BD · h = ½ · DC · h = ar(ΔACD)
Equal area is necessary for such a dissection, but it is not obvious that it is enough. Here it is, and the recipe is short:
- Let M be the midpoint of AB. Cut ΔABD along MD.
- Rotate the piece MBD through 180° about M. Then B lands on A, and because MD ∥ AC and MD = ½AC (midpoint theorem), the piece lands exactly inside ΔACD.
- The two pieces now cover ΔACD with no gap and no overlap.
Why it happens: ΔABD and ΔACD are generally not congruent — they have different shapes — so no single rigid motion carries one onto the other. But cutting frees us: a rigid motion of a part can rearrange the shape while preserving area. Area is what survives cutting and rearranging; congruence is much stronger.
Did you know? The general fact — any two polygons of equal area can be cut into finitely many pieces that reassemble into each other — is the Wallace–Bolyai–Gerwien theorem. It is a special feature of the plane: in three dimensions the corresponding statement is false (Dehn, 1900).