The Butterfly Model: Find the Wings in 10 Seconds

Step by step

1. A trapezoid: top and bottom are parallel
   Here's a trapezoid. Its top and bottom are parallel.
2. Draw both diagonals → a butterfly
   Draw both diagonals, from corner to corner. It looks like a butterfly!
3. Top = 4, bottom = 9. Each wing = ?
   The top is four. The bottom is nine. How big is each wing?
4. Surprise: the two wings are always equal
   First, a surprise. The two wings are always the same size. Here's why.
5. Same base, same height → same area
   Look at this pink triangle. Now slide its top corner along the top edge. Its base stays the same, and so does its height. So its area never changes.
6. Take away the bottom → the wings are equal
   Both pink triangles include the bottom piece. Take it away, and each one leaves a wing. So the two wings are equal!
7. Share a corner → compare like the bases
   Next, a trick from our bird's head video. Two triangles that share a corner, with bases on one line, compare like their bases.
8. Top vs. wing: same corner, bases on the diagonal
   The top and the right wing share this corner. Their bases are these two parts of the diagonal.
9. Wing vs. bottom: same two bases
   The left wing and the bottom share this corner, with the same two bases.
10. top : wing = wing : bottom
    So the top compares to a wing, the same way a wing compares to the bottom.
11. wing × wing = 4 × 9 = 36 → wing = 6
    That means wing times wing equals top times bottom. Four times nine is thirty-six, and six times six is thirty-six. So each wing is six!
12. Wings equal · wing × wing = top × bottom
    That's the butterfly model. The wings are equal, and wing times wing equals top times bottom.