Bhaskara II Acharya/Lilavati/Chapter VI/150
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Bhaskara II Acharya: Lilavati Chapter $\text {VI}$: Plane Figure: $150$
- A snake's hole is at the foot of a pillar which is $15$ cubits high and a peacock is perched on its summit.
- Seeing a snake, at a distance of thrice the pillar's height, gliding towards his hole, he pounces obliquely upon him.
- Say quickly at how many cubits from the snake's hole do they meet, both proceeding an equal distance?
Solution
They meet $25$ cubits away from the pillar.
Proof
Let $y$ be the distance away from the snake's hole where they meet.
Let the peacock and snake move $x$ cubits.
Then:
- $y = 45 = x$
The pillar, the ground and the flight of the peacock form a right triangle:
- with legs of $15$ and $45 - x$
- with hypotenuse $x$.
Hence:
\(\ds 15^2 + \paren {45 - x}^2\) | \(=\) | \(\ds x^2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 15^2 + 45^2 - 2 \times 45 x + x^2\) | \(=\) | \(\ds x^2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 45 \paren {5 + 45}\) | \(=\) | \(\ds 45 \paren {2 x}\) | simplifying and extracting factors for convenience | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds 25\) | simplifying |
It can be noted that the right triangle in question is the classic $\text {3-4-5}$ triangle expanded $5$ times.
$\blacksquare$
Sources
- 1816: John Taylor: Lilawati: or A Treatise on Arithmetic and Geometry by Bhascara Acharya: Part $\text {II}$: Chap. $\text {I}$: Of Geometrical Operations
- 1817: Henry Thomas Colebrooke: Algebra, with Arithmetic and Mensuration: Arithmetic (Lilavati): Chapter $\text {VI}$: Plane Figure: $150$
- 1893: Haran Chandra Banerji: Colebrooke's Translation of the Lilavati: Chapter $\text {VI}$: Plane Figure: $150$
- 1976: Howard Eves: Introduction to the History of Mathematics (4th ed.)
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Indian Puzzles: $56$