How to Measure the Width of a River or Pond (Similar Triangles)

Step by step

1. How wide is the pond? No tape across the water
   How wide is this pond? You can't stretch a tape across the water.
2. Two straight paths cross at a stake · short side parallel to the pond
   So walk two straight paths that cross at this stake. Then mark a short side, parallel to the pond.
3. 40 m to the stake · 10 m past it · short side 15 m
   This path is forty meters to the stake, and ten more past it. The short side is fifteen meters.
4. "The hourglass model": two triangles tip to tip
   In China, this is called the hourglass model, because the two triangles meet tip to tip, like an hourglass.
5. Way 1: turn the small triangle half a turn
   Way one: turn the small triangle half a turn around the stake.
6. Its sides land on the same paths · short side still parallel
   Each path is a straight line through the stake, so the turned sides land right on the same paths. And the short side is still parallel to the pond.
7. Zoom from the stake · 40 = 4 × 10 → zoom by 4
   Now zoom in from the stake, like zooming a photo. Forty is four times ten, so zoom by four.
8. The corner lands on the edge · the short side lands along the pond
   This corner lands exactly on the pond's edge. The short side stays parallel, so it lands right along the pond.
9. Every side × 4 → 15 × 4 = 60 m
   Zooming by four makes every side four times longer. So the pond is fifteen times four: sixty meters wide!
10. Way 2: the school way, with angles
    Here's a second way to see it: the school way, with angles.
11. Opposite angles at the stake are equal
    The two angles at the stake are opposite each other. Half a turn swaps them, so they're equal.
12. Parallel lines → equal Z angles
    The pond side and the short side are parallel. Parallel lines make equal Z angles.
13. All three add to 180° → the last pair matches too
    The last angles must match too, because all three add up to one hundred eighty degrees.
14. Same angles → same shape · big ÷ small is the same for every pair
    Same angles means same shape, just a different size. So big divided by small is the same for every pair of sides.
15. Careful: it's flipped · left path ↔ right path
    Careful: the small triangle is flipped. The path on the left matches the path on the right.
16. 40 ÷ 10 = 4 → 15 × 4 = 60 m
    Forty divided by ten is four. So the pond is fifteen times four: sixty meters again.
17. Hourglass: big ÷ small is the same for every matching pair
    So in an hourglass, big divided by small is the same for every matching pair of sides.
18. On a test: vertical angles · alternate interior angles → similar (AA) → × 4
    On a test, write it like this. Name the points. The angles at the stake are vertical angles. The Z angles are alternate interior angles. So the triangles are similar. Matching sides grow by forty over ten: four. The pond is sixty meters wide.