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GEOMETRYCoordinate GeometryMathematics Calculator

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Endpoint from Midpoint

Given endpoint P₁ and midpoint M, the other endpoint is P₂ = 2M − P₁. So x₂ = 2xₘ−x₁, y₂ = 2yₘ−y₁. M is the midpoint of P₁P₂, so P₂ is the reflection of P₁ across M.

Concept Fundamentals

P₂ = 2M − P₁

Formula

2xₘ − x₁

x₂

2yₘ − y₁

y₂

P₂ = reflection of P₁ across M

Reflection

Find EndpointEnter P₁ and midpoint M

Why This Mathematical Concept Matters

Why: Finding the unknown endpoint when you know one endpoint and the midpoint is common in geometry, graphics (symmetry), and navigation. The formula follows from M = (P₁+P₂)/2.

How: Solve M = (P₁+P₂)/2 for P₂: P₂ = 2M − P₁. Component-wise: x₂ = 2xₘ−x₁, y₂ = 2yₘ−y₁. M is the midpoint, so P₂ is the reflection of P₁ about M.

●P₂ is the midpoint of P₁ and its reflection across M.
●Same formula works in 3D: z₂ = 2zₘ−z₁.
●If P₁ = M, then P₂ = M (degenerate segment).

Sources:Khan Academy — MidpointWolfram MathWorld — Midpoint

Sample Examples — Click to Load & Calculate

Known Endpoint (P₁)

x₁

y₁

Midpoint (M)

xₘ

yₘ

⚠️For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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P₂ = 2M − P₁ given P₁ and midpoint M.

— Coordinate Geometry

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P₂ is reflection of P₁ across M.

— Property

Key Takeaways

• The midpoint is the average of the two endpoints: $M = \\frac{P_1 + P_2}{2}$
• To find the unknown endpoint: $P_2 = 2M - P_1$ (solve for P₂ from the midpoint formula)
• In coordinates: $x_2 = 2x_m - x_1$ and $y_2 = 2y_m - y_1$
• The midpoint bisects the segment — it is equidistant from both endpoints
• Finding the endpoint is equivalent to reflecting P₁ across M

Did You Know?

Reflection

Finding the endpoint P₂ given P₁ and midpoint M is the same as reflecting point P₁ across point M. The midpoint is the center of symmetry.

Vector Form

In vector notation: P₂ = 2M - P₁. This is a common operation in computer graphics and physics for point reflection.

Circle Diameter

If you know one endpoint of a diameter and the center (midpoint), you can find the other endpoint — essential for circle equations.

Segment Extension

Extending a segment beyond one endpoint: if you have P₁ and M, P₂ is the point that makes M the midpoint of P₁P₂.

Medians

In a triangle, the midpoint of each side connects to the opposite vertex to form a median. The centroid is at the average of the three vertices.

GPS & Navigation

Finding meeting points between two locations often uses midpoint logic. The endpoint formula extends this when the "center" is known.

Understanding the Endpoint Formula

The midpoint of a segment with endpoints $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is:

M = \\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right)

Solving for $P_2$ when $P_1$ and $M$ are known:

x_2 = 2x_m - x_1, \\quad y_2 = 2y_m - y_1 \\quad \\text{or} \\quad \\vec{P_2} = 2\\vec{M} - \\vec{P_1}

Expert Tips

Double and Subtract

Remember: P₂ = 2M - P₁. Double the midpoint coordinates, then subtract the known endpoint.

Verify with Distance

The distance from P₁ to M should equal the distance from M to P₂. Use this to check your answer.

Extend to 3D

For 3D: z₂ = 2zₘ - z₁. Same formula applies to each coordinate.

Collinearity Check

P₁, M, and P₂ are always collinear. M lies on the segment P₁P₂ and divides it in ratio 1:1.

Frequently Asked Questions

What is the endpoint formula?

Given endpoint P₁(x₁,y₁) and midpoint M(xₘ,yₘ), the other endpoint is P₂(2xₘ-x₁, 2yₘ-y₁). In vector form: P₂ = 2M - P₁.

How is this derived from the midpoint formula?

The midpoint M = (P₁ + P₂)/2. Multiplying by 2: 2M = P₁ + P₂. So P₂ = 2M - P₁.

Can I use this for 3D coordinates?

Yes. For 3D points, use z₂ = 2zₘ - z₁ in addition to the x and y formulas.

What if the midpoint equals the known endpoint?

Then P₂ = P₁. The segment has zero length — both endpoints coincide with the midpoint.

How do I find the midpoint from two endpoints?

Use M = ((x₁+x₂)/2, (y₁+y₂)/2). This calculator does the reverse: given one endpoint and the midpoint, find the other.

Is this related to point reflection?

Yes. Finding P₂ such that M is the midpoint of P₁P₂ is equivalent to reflecting P₁ across M to get P₂.

When is this used in practice?

In geometry (finding vertices), computer graphics (reflections, symmetry), navigation (meeting points), and physics (center of mass, symmetry).

How to Use This Calculator

Enter the known endpoint P₁ (x₁, y₁) and the midpoint M (xₘ, yₘ).
Click a sample example to auto-fill and calculate, or enter your own values.
Click "Calculate" to find the other endpoint P₂.
Review the visualization showing P₁, M, and P₂ on the coordinate plane.
Check the step-by-step solution for the derivation.
Copy results to share or paste into assignments.

Note: This calculator uses the formula P₂ = 2M - P₁. Results are suitable for educational purposes, homework, and professional calculations.

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