Question

What is the Endpoint Formula?

Answer 1

Hint: Endpoint formula to find the coordinates of end points of a line segment such that the midpoint of the line segment connecting the two endpoints is known and one of the endpoints coordinate is known. We have to consider two endpoints $A(x_1,y_1)$ and $B(x_2,y_2)$ ,and the midpoint $M(x,y)$ . Of these, we have to assume $A(x_1,y_1)$ and $M(x,y)$ are known. We will use the midpoint formula $M(x,y)=({\displaystyle \frac{x_1+x_2}{2}},{\displaystyle \frac{y_1+y_2}{2}})$ . Next, we have to equate the coordinates and find the value of $(x_2,y_2)$ .

Complete step by step solution:
We use an endpoint formula to find the coordinates of end points of a line segment such that the midpoint of the line segment connecting the two endpoints is known and one of the endpoints coordinate is known. Let us consider a line segment with endpoints $A(x_1,y_1)$ and $B(x_2,y_2)$ . $M(x,y)$ is the midpoint. This is illustrated below.

Let us suppose that the midpoint and the one of the endpoint, that is, $M(x,y)$ and $A(x_1,y_1)$ , are known. We can write the midpoint formula for the midpoint M of a line segment joining the endpoints $A(x_1,y_1)$ and $B(x_2,y_2)$ as
$M(x,y)=({\displaystyle \frac{x_1+x_2}{2}},{\displaystyle \frac{y_1+y_2}{2}})$
Let us equate the coordinates. Let us first equate x coordinates.
$x={\displaystyle \frac{x_1+x_2}{2}}$
Let us find $x_2$ from the above equation. We will first take 2 from LHS to RHS.
$\Rightarrow 2x=x_1+x_2$
Let us take $x_1$ to LHS.
$\Rightarrow x_2=2x-x_1...\left(i\right)$
Now, we have to find $y_2$ by equating the y coordinates.
$y={\displaystyle \frac{y_1+y_2}{2}}$
We will first take 2 from LHS to RHS.
$\Rightarrow 2y=y_1+y_2$
Let us take $y_1$ to LHS.
$\Rightarrow y_2=2y-y_1$
Hence, the endpoint formula that is used to find the endpoint is given as $(2x-x_1,2y-y_1)$ .

Note: Students must know what endpoints and midpoints are. Endpoint is any of the two furthest points on a line segment. Midpoint is the middle point of a line segment. It is equidistant from both endpoints. Students must know the midpoint formula to derive the endpoint formula.