Shifts and the Baudhāyana–Pythagoras theorem
d = √((x₂ − x₁)² + (y₂ − y₁)²)
Across the corner
Go from A to D: 4 units right and 3 units down. The straight line AD cuts across the corner.
How long is the straight line AD?
What this lesson covers
The idea
By the Baudhāyana–Pythagoras theorem applied to the shifts along the two axes, the distance between (x₁, y₁) and (x₂, y₂) is √((x₂ – x₁)² + (y₂ – y₁)²); the signs of the shifts make no difference.
Across the corner
Go from A to D: 4 units right and 3 units down. The straight line AD cuts across the corner.
How long is the straight line AD?
- 7 units (4 + 3)
- 5 units
- 12 units (4 × 3)
Shifts and squares
Move B anywhere. The two shifts form a right triangle with AB as its longest side. Watch the numbers below the picture.
The distance formula
By the Baudhāyana–Pythagoras theorem applied to the shifts along the two axes, the distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²). The signs of the shifts make no difference.
The shifts are x₂ − x₁ and y₂ − y₁ (second point minus first). Squaring wipes out any minus sign. A′ and D′ are A and D reflected in the y-axis, and reflection does not change lengths. Another pair from the book: D (7, 1) and M (9, 6) give √(2² + 5²) = √29.
- Pair | Shifts | Distance
- A (3, 4), D (7, 1) | 4 and −3 | √(16 + 9) = 5
- A′ (−3, 4), D′ (−7, 1) | −4 and −3 | √(16 + 9) = 5
Notes
The distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²). The signs of the shifts make no difference.
Check yourself
What is the distance between (0, 0) and (6, 8)?
Answer: 10
√(6² + 8²) = √(36 + 64) = √100 = 10.
What is the distance between (−2, 1) and (2, 4)?
Answer: 5
√(4² + 3²) = √(16 + 9) = √25 = 5.
A (3, 4) and D (7, 1) are 5 units apart. Reflect both points in the y-axis to get A′ (−3, 4) and D′ (−7, 1). How far apart are A′ and D′?
A (1, 1) and B (3, 6). What is (x₂ − x₁)² + (y₂ − y₁)²? (The distance AB is the square root of it.)
Answer: 29
2² + 5² = 4 + 25 = 29, so AB = √29.
- 5 units — correct. Yes. The shifts are −4 and −3. Their squares are still 16 and 9, so the distance is √25 = 5.
- 7 units. The shifts are 4 and 3 in size. Square them, add: 16 + 9 = 25, and √25 = 5.
- −5 units. The signs of the shifts make no difference, and the distance comes out as the length 5.