SamCalculator

Distance Geometry Toolkit

Ten coordinate-geometry mini-tools on a single live dashboard.

AA′BMθ

Coordinate-geometry transformations

d = √Δx² + Δy², M = ((x₁+x₂)/2, (y₁+y₂)/2)

Live coordinate-geometry dashboard

Point A

Point B

Distance

10

Midpoint

(5, 7)

Slope

4/3

Angle

53.13°

Bearing A→B

36.9° NE

y-intercept

0.3333

Direction

Quadrant 1

Δx = 6Δy = 8P₁ (2, 3)P₂ (8, 11)

Line equation

y = 4/3x + 0.333333

4x − 3y + 1 = 0

Point reflection

Reflect Point A across a chosen line or the origin. The original point's distance from the line is mirrored on the opposite side.

Original A

(2, 3)

Reflected A′

(2, -3)

Translation by a vector

Shift Point A by the vector ⟨tx, ty⟩. Distance and shape are preserved — only the position changes.

A

(2, 3)

A + ⟨tx, ty⟩

(5, 1)

Rotation around a centre

Rotate Point A by an angle θ (degrees, counter-clockwise) around an arbitrary centre.

A

(2, 3)

Rotated A′

(-3, 2)

Vector magnitude

Length of a vector in any number of dimensions: ‖v‖ = √Σ vᵢ².

‖v‖

5

What is the Distance Geometry Toolkit?

A single page that bundles the most common short coordinate-geometry calculations: distance between points, midpoint, slope, line equation, bearing, vector magnitude, reflection across an axis or line, translation, rotation around a centre, and a live dashboard tying everything together. Each tool takes a small set of inputs and returns a single focused answer, so you can quickly chain them when working through a problem.

The toolkit is designed for homework, quick sanity checks, and exploratory geometry. Slide the points around and watch the distance, midpoint, slope, line equation, and bearing all update in real time — it is much easier to build intuition for what 'slope = −2' feels like when you can drag a point and see the line tilt rather than re-deriving the slope each time.

How the toolkit works

Pick a tool

Each card holds one focused calculation. Inputs sit at the top, the result shows immediately below — no separate Calculate button for the simple tools.

Use the dashboard

The interactive dashboard at the top shows two points whose coordinates you can edit; every cared-about value (distance, slope, midpoint, bearing, line equation, vector magnitude) updates live.

Read the step-by-step

Every tool exposes its own short solution panel so you can see exactly how the answer was derived.

Copy or share

The Copy and Share buttons capture every result on the page into a tab-separated block ready to paste into a notebook or chat.

Formulas used by the toolkit

2-D distance

d = √[(x₂−x₁)² + (y₂−y₁)²]

Straight-line distance between two points.

Midpoint

M = ((x₁+x₂)/2, (y₁+y₂)/2)

Exact halfway point between two coordinates.

Slope

m = (y₂−y₁) / (x₂−x₁)

Rise over run; undefined when x₁ = x₂.

Bearing

θ = atan2(Δx, Δy) (mod 360°)

Angle measured clockwise from north.

Reflection across y = x

(x, y) → (y, x)

Swap the coordinates.

Rotation around origin

x' = x cos θ − y sin θ, y' = x sin θ + y cos θ

Multiply by the rotation matrix.

Translation

(x, y) → (x + tx, y + ty)

Shift every point by the vector ⟨tx, ty⟩.

Vector magnitude

‖v‖ = √Σ vᵢ²

Length of the vector in any number of dimensions.

Where these tools come up

Coursework

GCSE / A-level / high-school geometry homework, the AP Precalculus Coordinate Geometry unit, and college-level Linear Algebra warm-ups.

CAD and modelling

Reflecting a sketch across an axis, translating a feature by a vector, rotating a part by an angle — the same operations powering professional CAD tooling.

Robotics and graphics

Computing direction, rotating points around a pivot, and translating geometry between coordinate frames are everyday operations in robotics control loops and 3-D graphics pipelines.

Navigation

Bearing and distance between two coordinates show up in any mapping or wayfinding task — from orienteering to drone telemetry.

Why use the toolkit?

Each of these formulas is short, but doing many of them in sequence on paper is error-prone and slow. Having them collected into one tabbed page with sliders and a shared coordinate canvas makes it much faster to verify a multi-step problem and to build intuition by watching values update as you move the points.

Frequently Asked Questions

Ten short tools covering the most common coordinate-geometry calculations: distance between points, midpoint, slope, line equation, bearing, vector magnitude, point reflection across an axis or line, translation by a vector, rotation around a centre, and a live coordinate dashboard tying the lot together.

Flip the sign of the y-coordinate: (x, y) → (x, −y). Across the y-axis becomes (−x, y), across the origin becomes (−x, −y), across y = x becomes (y, x), and across y = −x becomes (−y, −x). The Reflection tool inside the toolkit applies whichever rule you select.

A rigid shift of every point by the same vector ⟨tx, ty⟩. The new coordinates are (x + tx, y + ty). Translations preserve distances and angles — they only change position, not shape or orientation.

Multiply by the rotation matrix: x' = x cos θ − y sin θ, y' = x sin θ + y cos θ. The Rotation tool also accepts an arbitrary centre — it translates the centre to the origin, rotates, then translates back.

An angle measured clockwise from north (the positive y-axis) to the direction of a target point, ranging from 0° to 360°. It is the navigational convention, which differs from the trigonometric angle measured counter-clockwise from the positive x-axis by 90° plus a sign flip.

Use point-slope form: y − y₁ = m(x − x₁). Distribute m and add y₁ to get slope-intercept form y = mx + (y₁ − m·x₁), or move every term to one side to get standard form Ax + By + C = 0.

The length of the vector: ‖v‖ = √(v₁² + v₂² + … + vₙ²). It tells you 'how big' the vector is regardless of direction. Dividing every component by the magnitude gives the unit vector — a vector of length 1 pointing the same way.

Because the two points share the same x-coordinate (Δx = 0), so the line between them is vertical. Vertical lines have no finite slope; their equation is written x = c rather than y = mx + b.

Every input is bound to a slider; moving a slider updates the point coordinates, which in turn recompute the distance, midpoint, slope, bearing, and line equation instantly. You can also drag the points on the canvas directly.

Yes. The Copy and Share buttons at the bottom of the result panel capture every computed value — distance, slope, midpoint, line equation, bearing — into a tab-separated block suitable for pasting into a notebook, spreadsheet, or chat.

Related Calculators

More coordinate-geometry, trig, and unit tools that pair with distance calculations.