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 $A\left( {{x}_{1}},{{y}_{1}} \right)$ and $B\left( {{x}_{2}},{{y}_{2}} \right)$ ,and the midpoint $M\left( x,y \right)$ . Of these, we have to assume $A\left( {{x}_{1}},{{y}_{1}} \right)$ and $M\left( x,y \right)$ are known. We will use the midpoint formula $M\left( x,y \right)=\left( \dfrac{{{x}_{1}}+{{x}_{2}}}{2},\dfrac{{{y}_{1}}+{{y}_{2}}}{2} \right)$ . Next, we have to equate the coordinates and find the value of $\left( {{x}_{2}},{{y}_{2}} \right)$ .

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 $A\left( {{x}_{1}},{{y}_{1}} \right)$ and $B\left( {{x}_{2}},{{y}_{2}} \right)$ . $M\left( x,y \right)$ is the midpoint. This is illustrated below.

Let us suppose that the midpoint and the one of the endpoint, that is, $M\left( x,y \right)$ and $A\left( {{x}_{1}},{{y}_{1}} \right)$ , are known. We can write the midpoint formula for the midpoint M of a line segment joining the endpoints $A\left( {{x}_{1}},{{y}_{1}} \right)$ and $B\left( {{x}_{2}},{{y}_{2}} \right)$ as
$M\left( x,y \right)=\left( \dfrac{{{x}_{1}}+{{x}_{2}}}{2},\dfrac{{{y}_{1}}+{{y}_{2}}}{2} \right)$
Let us equate the coordinates. Let us first equate x coordinates.
$x=\dfrac{{{x}_{1}}+{{x}_{2}}}{2}$
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.
$y=\dfrac{{{y}_{1}}+{{y}_{2}}}{2}$
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 $\left( 2x-{{x}_{1}},2y-{{y}_{1}} \right)$ .

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.