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.

What is the Distance formula?

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

When do you use Distance?

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