Distance in Two Dimensions

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Key Ideas Easy Medium Hard

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

What to Remember

Formula

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

Squares first

Subtract each pair of coordinates, square, then add.

Simplify the root

Take square factors out of the radicand. A perfect square becomes a whole number.

Same coordinate

If one coordinate matches, that squared difference is 0. The distance is the absolute difference of the other coordinate.

Easy

The distance is a whole number.

Example

Find the exact distance between (1, 2) and (6, 14).

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (6 − 1)² + (14 − 2)²

d = square root of 25 + 144

d = square root of 169

d = 13

Answer: 13

Find the exact distance between (0, 0) and (6, 8).

Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (6 − 0)² + (8 − 0)²

d = square root of 36 + 64

d = square root of 100

d = 10

Answer: 10

Find the exact distance between (1, 2) and (4, 6).

Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (4 − 1)² + (6 − 2)²

d = square root of 9 + 16

d = square root of 25

d = 5

Answer: 5

Find the exact distance between (−2, 1) and (3, 13).

Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (3 − (−2))² + (13 − 1)²

d = square root of 25 + 144

d = square root of 169

d = 13

Answer: 13

Find the exact distance between (2, −1) and (6, 2).

Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (6 − 2)² + (2 − (−1))²

d = square root of 16 + 9

d = square root of 25

d = 5

Answer: 5

Find the exact distance between (−4, 2) and (4, 8).

Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (4 − (−4))² + (8 − 2)²

d = square root of 64 + 36

d = square root of 100

d = 10

Answer: 10

Medium

A horizontal or vertical segment, or a larger triple.

Example

Find the exact distance between (1, 1) and (4, 5).

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (4 − 1)² + (5 − 1)²

d = square root of 9 + 16

d = square root of 25

d = 5

Answer: 5

Find the exact distance between (0, 3) and (5, 3).

Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (5 − 0)² + (3 − 3)²

d = square root of 25 + 0

d = square root of 25

d = 5

Answer: 5

Find the exact distance between (3, −2) and (3, 6).

Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (3 − 3)² + (6 − (−2))²

d = square root of 0 + 64

d = square root of 64

d = 8

Answer: 8

Find the exact distance between (−1, 4) and (7, −2).

Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (7 − (−1))² + (−2 − 4)²

d = square root of 64 + 36

d = square root of 100

d = 10

Answer: 10

Find the exact distance between (2, 5) and (8, 8).

Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (8 − 2)² + (8 − 5)²

d = square root of 36 + 9

d = square root of 45

d = 3square root of 5

Answer: 3square root of 5

Find the exact distance between (4, 0) and (0, 4).

Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (0 − 4)² + (4 − 0)²

d = square root of 16 + 16

d = square root of 32

d = 4square root of 2

Answer: 4square root of 2

Hard

Simplify a root that is not a perfect square.

Example

Find the exact distance between (1, −3) and (6, 9).

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (6 − 1)² + (9 − (−3))²

d = square root of 25 + 144

d = square root of 169

d = 13

Answer: 13

Find the exact distance between (−5, −1) and (1, 7).

Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (1 − (−5))² + (7 − (−1))²

d = square root of 36 + 64

d = square root of 100

d = 10

Answer: 10

Find the exact distance between (2, 2) and (7, 14).

Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (7 − 2)² + (14 − 2)²

d = square root of 25 + 144

d = square root of 169

d = 13

Answer: 13

Find the exact distance between (−3, 4) and (5, −2).

Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (5 − (−3))² + (−2 − 4)²

d = square root of 64 + 36

d = square root of 100

d = 10

Answer: 10

Find the exact distance between (1, −2) and (9, 4).

Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (9 − 1)² + (4 − (−2))²

d = square root of 64 + 36

d = square root of 100

d = 10

Answer: 10

Find the exact distance between (−6, 3) and (2, −3).

Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (2 − (−6))² + (−3 − 3)²

d = square root of 64 + 36

d = square root of 100

d = 10

Answer: 10

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