Finding the Endpoint of a Line Segment Using the Midpoint Formula

In this article, we will explore how to find the endpoint of a line segment when you have one endpoint and the midpoint. We'll walk through a detailed example to understand the process step by step. By the end, you'll be able to solve similar problems using the midpoint formula.

Understanding the Midpoint Formula

The midpoint of a line segment with endpoints ( (x_1, y_1) ) and ( (x_2, y_2) ) is given by the formula:

(x1 x22,mfrac>y1 y22)

Solving for the Endpoint

Let's consider a specific example where we know one endpoint is ( (9, -10) ) and the midpoint is ( (4, 8) ). We need to find the coordinates of the unknown endpoint ( (x_2, y_2) ).

Setting Up Equations

Using the midpoint formula, we can set up two equations, one for the ( x )-coordinate and one for the ( y )-coordinate. For the ( x )-coordinate:

x1 x2 2 cdot 4

Simplifying, we get:

x1 x2 8

Substituting the known ( x )-coordinate of the first endpoint, ( x_1 9 ), we have:

9 x2 8

Solving for ( x_2 ):

x2 8 - 9 -1

For the ( y )-coordinate:

y1 y2 2 cdot 8

Simplifying, we get:

y1 y2 16

Substituting the known ( y )-coordinate of the first endpoint, ( y_1 -10 ), we have:

-10 y2 16

Solving for ( y_2 ):

y2 16 10 26

Conclusion

The coordinates of the unknown endpoint are ( (-1, 26) ). Therefore, the endpoint of the line segment is ( (-1, 26) ).

Additional Examples

Let's consider another example where the other endpoint is assumed to be ( (x, y) ), and the midpoint is given as ( (4, 8) ), with the known endpoint ( (9, -10) ). Given the midpoint formula,

x1 x2 2 cdot 4

or,

x1 x2 8

Substituting ( x_1 9 ), we get:

9 x2 8

Solving for ( x ), we find:

x -1

Similarly, for the ( y )-coordinate,

y1 y2 2 cdot 8

or,

y1 y2 16

Substituting ( y_1 -10 ), we get:

-10 y2 16

Solving for ( y ), we find:

y 26

Therefore, the coordinates of the other endpoint are ( (-1, 26) ).

Conclusion

By leveraging the midpoint formula, we can accurately find the coordinates of an unknown endpoint given one endpoint and the midpoint. This method is useful in various fields, including geometry, cartography, and data analysis. Understanding and applying the midpoint formula is an essential skill that can help solve a wide range of problems efficiently.