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← The Baudhayana-Pythagoras Theorem

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Lesson 12 of 16

The hidden depth of a pond

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9 min

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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.

Try the lab to finish this lesson.