Distance

Geometry
definition

Also known as: length between points, distance formula, how far apart, distance

Grade 9-12

View on concept map

The length of the shortest path between two points, always a non-negative real number. Fundamental measurement in geometry, physics, and navigation.

Definition

The length of the shortest path between two points, always a non-negative real number.

💡 Intuition

'As the crow flies'—the straight-line separation between two locations.

🎯 Core Idea

Distance measures separation between two points; it is always non-negative and zero only when the points coincide.

Example

Distance between (0,0) and (3,4) is \sqrt{3^2 + 4^2} = 5

Formula

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Notation

d(A,B) or |AB| denotes the distance between points A and B

🌟 Why It Matters

Fundamental measurement in geometry, physics, and navigation.

💭 Hint When Stuck

Draw a right triangle between the two points. The horizontal and vertical legs give you the values to plug into the Pythagorean theorem.

Formal View

Euclidean distance: d(P,Q) = \|P - Q\| = \sqrt{\sum_{i=1}^n (p_i - q_i)^2} for P,Q \in \mathbb{R}^n; satisfies metric axioms: d(P,Q) \geq 0, d(P,Q) = 0 \iff P = Q, d(P,Q) = d(Q,P), d(P,R) \leq d(P,Q) + d(Q,R)

🚧 Common Stuck Point

Distance on a plane uses the Pythagorean theorem (distance formula).

⚠️ Common Mistakes

  • Computing (x_2 - x_1) + (y_2 - y_1) instead of using the Pythagorean theorem — that gives the taxicab distance, not straight-line distance
  • Forgetting to take the square root after computing (\Delta x)^2 + (\Delta y)^2
  • Reporting a negative distance — distance is always non-negative

Frequently Asked Questions

What is Distance in Math?

The length of the shortest path between two points, always a non-negative real number.

Why is Distance important?

Fundamental measurement in geometry, physics, and navigation.

What do students usually get wrong about Distance?

Distance on a plane uses the Pythagorean theorem (distance formula).

What should I learn before Distance?

Before studying Distance, you should understand: pythagorean theorem.

Prerequisites

pythagorean theorem

Next Steps

distance formula

How Distance Connects to Other Ideas

To understand distance, you should first be comfortable with pythagorean theorem. Once you have a solid grasp of distance, you can move on to distance formula.

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