Endpoint Calculator

How to Calculate the Endpoint of a Line Segment

Finding the endpoint of a line segment when given one endpoint and the midpoint is a common problem in coordinate geometry. Here's a comprehensive guide on how to perform this calculation:

Endpoint Formula

The formula to find the coordinates of the unknown endpoint (x₂, y₂) given one endpoint (x₁, y₁) and the midpoint (xₘ, yₘ) is:

\[ x_2 = 2x_m - x_1 \quad \text{and} \quad y_2 = 2y_m - y_1 \]

Where:

• (x₁, y₁) are the coordinates of the known endpoint
• (xₘ, yₘ) are the coordinates of the midpoint
• (x₂, y₂) are the coordinates of the unknown endpoint we're calculating

Calculation Steps

1. Identify the coordinates of the known endpoint (x₁, y₁) and the midpoint (xₘ, yₘ)
2. Use the formula x₂ = 2xₘ - x₁ to calculate the x-coordinate of the unknown endpoint
3. Use the formula y₂ = 2yₘ - y₁ to calculate the y-coordinate of the unknown endpoint
4. Combine the calculated x₂ and y₂ to get the coordinates of the unknown endpoint (x₂, y₂)

Example: Find the endpoint given (2, 3) and midpoint (4, 5)

1. Known endpoint (x₁, y₁) = (2, 3) and midpoint (xₘ, yₘ) = (4, 5)
2. Calculate x₂: \[ x_2 = 2(4) - 2 = 8 - 2 = 6 \]
3. Calculate y₂: \[ y_2 = 2(5) - 3 = 10 - 3 = 7 \]
4. The unknown endpoint is (6, 7)

Visual Representation

A visual representation can help understand the concept better. Here's a diagram showing the line segment with endpoints (2, 3) and (6, 7), and midpoint (4, 5):

This diagram illustrates how the known endpoint (2, 3), midpoint (4, 5), and calculated endpoint (6, 7) form a line segment.