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The Baudhayana-Pythagoras Theorem

The hidden depth of a pond

A made-up lotus stem stands 10 cm above a pond. A breeze (a light wind) bends it. The flower touches the water 50 cm from where the stem stood. How deep is the pond?

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The stem stands straight. It is d under the water and 10 cm above.
Step 1 of 4
  1. The stem stands straight. It is d under the water and 10 cm above.
  2. The breeze bends it. The flower touches the water.
  3. The bent stem is the long side of a right triangle. The legs are d and 50.
  4. d² + 50² = (d + 10)². So d = 120 cm.

The rule

The pond is 120 cm deep.

You cannot measure the depth. Call the depth d. The stem keeps its length when it bends. So the long side is d plus the part above the water. The depth runs straight up from the root to the flat water. So it meets the water at a square corner.

Rule: The bent stem keeps its length. Write the sides of the right triangle with d, then solve.

The legs are d and 50. So d² + 50² = (d + 10)². Here 50² = 2,500 and (d + 10)² = d² + 20d + 100. So d² + 2,500 = d² + 20d + 100. Both sides hold d², so it cancels. That leaves 2,500 = 20d + 100. Subtract 100 from both sides: 20d = 2,400. Divide by 20: d = 120.

Check: 120² + 50² = 14,400 + 2,500 = 16,900 = 130².

Jump to lesson

This track

  1. 0 Start here
  2. 1 Naming the sides
  3. 2 A square on every side
  4. 3 The two smaller squares together
  5. 4 Only at a right angle
  6. 5 Finding the long side
  7. 6 Finding a shorter side
  8. 7 Testing a triangle without a protractor
  9. 8 Where it is used
  10. 9 Corner to corner of a square
  11. 10 The long side as a decimal
  12. 11 One triple makes many
  13. 12 The hidden depth of a pond
  14. 13 Checkpoint: roots, triples and a pond
  15. 14 The big picture
  16. 15 Track quiz

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