Endpoint Formula 

The endpoint formula is related to the midpoint formula. The point in the middle/center of the line joining two points (also known as endpoints) is called a midpoint. Given one endpoint and a midpoint, the other midpoint can be calculated using the midpoint formula. Let us explore the endpoint formula below.

What is Endpoint Formula?

The endpoint formula helps in determining the values of the endpoints in either a line segment or a ray. Endpoints are the endpoints of a line segment and one endpoint if it is a ray where both the line segment and the ray stop. The line does not extend any further from the endpoints. To calculate the endpoints we need to know the midpoint formula. The midpoint is the center or middle point of any line that lies in the center of the endpoints. 

Let M (\((x)_{m}\) , \((y)_{m}\)) be a midpoint for the line joining two endpoints A (\((x)_{1}\) , \((y)_{1}\)) and B (\((x)_{2}\) , \((y)_{2}\)). We can use the midpoint formula to solve for either of the endpoints. Given the coordinates of M and A, the coordinates of B can be calculated using the following formula: 

(From the midpoint formula)

\((x)_{m}\) = \( \dfrac{x_1 + x_2}{2} \),

\((y)_{m}\) = \( \dfrac{y_1 + y_2}{2} \)

\((x)_{2}\) = 2\((x)_{m}\) - \((x)_{1}\),

\((y)_{2}\) = 2\((y)_{m}\) - \((y)_{1}\) 

Thus, the endpoint formula is,

Endpoint formula of B(\((x)_{2}\), \((y)_{2}\)) = (2\((x)_{m}\) - \((x)_{1}\),  2\((y)_{m}\) - \((y)_{1}\))

Note: It is not recommended to learn this formula, rather just find the coordinates of B by just using the midpoint formula.

Endpoint Formula 

By solving the midpoint formula for the points x₂ and y₂, we get the endpoint formula, i.e.

Endpoint formula of B(\((x)_{2}\), \((y)_{2}\)) = (2\((x)_{m}\) - \((x)_{1}\),  2\((y)_{m}\) - \((y)_{1}\))

Examples Using Endpoint Formula

Example 1: M(3, 4) is the midpoint of the line joining points A(5, 2) and B(x, y). Find the coordinates of B using the endpoint formula.

Solution:

Given, 

\((x)_{m}\) = 3, \((x)_{1}\) = 5, \((y)_{m}\) = 4, \((y)_{1}\) = 2

Using the endpoint formula, 

\((x)_{2}\) = 2\((x)_{m}\) - \((x)_{1}\)

\((x)_{2}\) = 2 × 3 - 5

\((x)_{2}\) = 1

\((y)_{2}\) = 2\((y)_{m}\) - \((y)_{1}\) 

\((y)_{2}\) = 2 × 4 - 2

\((y)_{2}\) = 6

x = 1, y = 6

Therefore, the coordinates of B(x, y) = (1, 6), i.e. x = 1 and y = 6

Example 2:  C (7, 8) is the center of the circle having a radius = 5 units. A diameter is drawn on this circle, and one of its endpoints is (3, 5). Find the other endpoint of the diameter using the endpoint formula.

Solution:

Let E(x, y) be the other endpoint and C is the midpoint of a diameter. Given, 

\((x)_{m}\) = 7, \((x)_{1}\) = 3, \((y)_{m}\) = 8, \((y)_{1}\) = 5

Using the endpoint formula, 

\((x)_{2}\) = 2\((x)_{m}\) - \((x)_{1}\)

\((x)_{2}\) = 2 × 7 - 3

\((x)_{2}\)  = 11

\((y)_{2}\) = 2\((y)_{m}\) - \((y)_{1}\) 

\((y)_{2}\) = 2 × 8 - 5

\((y)_{2}\) = 11

x = 11, y = 11

Therefore, the coordinates of the other endpoint E(x, y) is (11, 11)

Example 3: P(5, 8) is the midpoint of the line joining points A(4, 3) and B(x, y). Find the coordinates of B using the endpoint formula.

Solution: 

Given, 

\((x)_{m}\) = 5 ,  \((x)_{1}\) = 4, \((y)_{m}\) = 8,  \((y)_{1}\)  = 3

Using the Endpoint Formula, 

\((x)_{2}\) = 2\((x)_{m}\) - \((x)_{1}\)

\((x)_{2}\) = 2 × 5 - 4 

\((x)_{2}\) = 6

\((y)_{2}\) = 2\((y)_{m}\) - \((y)_{1}\) 

\((y)_{2}\) = 2 × 8 - 3

\((y)_{2}\) = 13

x = 6, y = 13

Therefore, the coordinates of B(x, y) = (6, 13), i.e. x = 6 and y = 13