2. The Baudhāyana-Pythagoras Theorem

Part II, printed Chapter 28 topics1 h

Chapter 2 of NCERT Mathematics for Class 8 (Part II, printed Chapter 2): The Baudhāyana-Pythagoras Theorem. Three sections — Doubling and halving a square, The theorem, and the number it produces and Integer triples, and beyond. Eight videos, 1 h in all.

Worked answers to the 19 exercise questions in this chapter →

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Part 1

Doubling and halving a square

9 minDoubling a square: the diagonal is the constructionDraw a square of exactly twice the area. Not roughly twice — exactly. Baudhāyana set that problem down around 800 BCE.9 min · 12 partsहिंदी9 minHalving a square: the doubling construction run backwardsDraw a square of exactly half the area — and the thing you draw has to be a square. That last condition is the whole difficulty.9 min · 12 partsहिंदी

Part 2

The theorem, and the number it produces

10 minThe isosceles right triangle: why its hypotenuse must be a√2A right triangle with two sides of length one. How long is the third? A ruler will not settle it and you cannot look it up.10 min · 12 partsहिंदी9 minWhy √2 is neither a terminating decimal nor a fractionThe third side has a square of exactly 2. That is a complete answer — and it is a name, not a value.9 min · 12 partsहिंदी10 minCombining two different squares: a² + b² = c²Two equal squares combine into one, and a diagonal does the combining. Now drop the sameness and see what survives.10 min · 12 partsहिंदी

Part 3

Integer triples, and beyond

11 minBaudhāyana-Pythagoras triples, and how to make infinitely many3, 4, 5 — nine plus sixteen is twenty-five, so three whole numbers make a right triangle. There are infinitely many more, and a recipe for them.11 min · 12 partsहिंदी10 minFermat's Last Theorem: the same question one power upSquares split into two smaller squares over and over. Try it one power up and nobody found a single example in 350 years.10 min · 12 partsहिंदी10 minApplying the theorem: the lotus-in-the-lake problem from the LīlāvatīA lotus stands one unit clear of a lake. A breeze tips it until the flower touches the surface, three units away — how deep is the water?10 min · 12 partsहिंदी

What you will be able to do

  • Explain why doubling every side of a square multiplies its area by four rather than by two, and give the general reason in terms of side × side
  • Construct, inside a given square, a square of exactly half its area, by joining the midpoints of the four sides
  • Name the hypotenuse of a right triangle as the side facing the right angle
  • Bound √2 between 1 and 2 by comparing the areas of squares, and name 1 the lower bound and 2 the upper bound
  • State Baudhāyana's theorem for a right-angled triangle with sides a, b and hypotenuse c
  • Test whether a given integer triple satisfies a² + b² = c²
  • State the equation xⁿ + yⁿ = zⁿ and the conditions the chapter attaches to it: natural numbers, and an exponent above 2
  • Turn a problem stated in words and a picture into a labelled right triangle

What this chapter assumes you already know

  • A square's area is side × side, so scaling a side by a factor scales the area by that factor twice over
  • A square's two diagonals are equal, cross at their midpoints, meet at right angles, and each one bisects the two corner angles it passes through
  • Two triangles that agree in all three sides, or in two sides and the angle between them, are congruent, and congruent figures have equal area
  • Cutting a figure into non-overlapping pieces and rearranging them leaves its area unchanged
  • Reading a diagram in which some lines are drawn solid and others dotted, and knowing that a dotted line is still a real line of the construction
  • A square's diagonals are equal, bisect each other, and cross at right angles
  • Midpoint of a segment, and that joining midpoints of adjacent sides of a square gives a segment cutting off a right-angled corner triangle
  • Congruent triangles have equal area, and the two standard tests (three sides; two sides with the included angle)

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "Double the area means double the side."
  • "Then double the area means side times 1.5, or thereabouts."
  • "The tilted square is smaller — it looks pinched."
  • "The four triangles look the same, so they are."
  • "The extended sides pass through the corners because the drawing was made that way."
  • "Cutting and rearranging might lose a bit."
  • "This only works for a square."
  • "Half the side, half the area."