Question

How do you find the missing coordinate given one point A (2, 8) and the mid – point M (5, 4)?

Answer 1

Assume the point whose coordinates we need to find as $B\left( {{x}_{2}},{{y}_{2}} \right)$, coordinates of A as $A\left( {{x}_{1}},{{y}_{1}} \right)$ and the mid – point M as $M\left( x,y \right)$. Now, apply the mid – point formula for the coordinates x and y given as $x=\dfrac{{{x}_{1}}+{{x}_{2}}}{2}$ and $y=\dfrac{{{y}_{1}}+{{y}_{2}}}{2}$. Substitute all the given values and determine the coordinates of $\left( {{x}_{2}},{{y}_{2}} \right)$ to get the answer.

Complete step by step answer:
Here we have been provided with the coordinates of a point A and the coordinates of a mid – point M. we are asked to find the coordinates of the point such that M will be the mid – point of the line segment AB. We need to apply the mid – point formula to solve the question. Let us draw a diagram of the given situation.

Now, assuming the coordinates of coordinates of A as $A\left( {{x}_{1}},{{y}_{1}} \right)$, the mid – point M as $M\left( x,y \right)$ and the missing point B as $B\left( {{x}_{2}},{{y}_{2}} \right)$ we have the following data: -
$\Rightarrow {{x}_{1}}=2,{{y}_{1}}=8$ and $x=5,y=4$
We know that the coordinates of the mid – point of a line segment according to the above assumed coordinates is given by the mid – point formula. The x – coordinate is given as $x=\dfrac{{{x}_{1}}+{{x}_{2}}}{2}$ and the y – coordinate as $y=\dfrac{{{y}_{1}}+{{y}_{2}}}{2}$. Let us solve for them one by one.
(i) For x – coordinate we have,
$\Rightarrow x=\dfrac{{{x}_{1}}+{{x}_{2}}}{2}$
Substituting the given values in the above relation we get,
$\Rightarrow 5=\dfrac{2+{{x}_{2}}}{2}$
By cross – multiplication we get,
$\begin{aligned} & \Rightarrow 10=2+{{x}_{2}} \\\ & \Rightarrow 10-2={{x}_{2}} \\\ & \therefore {{x}_{2}}=8 \\\ \end{aligned}$
(i) For y – coordinate we have,
$y=\dfrac{{{y}_{1}}+{{y}_{2}}}{2}$
Substituting the given values in the above relation we get,
$\Rightarrow 4=\dfrac{2+{{y}_{2}}}{2}$
By cross – multiplication we get,
$\begin{aligned} & \Rightarrow 8=2+{{y}_{2}} \\\ & \Rightarrow 8-2={{y}_{2}} \\\ & \therefore {{y}_{2}}=6 \\\ \end{aligned}$
Hence the required coordinates of the point B is B (8, 6).

Note: Note that the mid – point formula is a special case of the section formula in which a point divides the line segment joining two points internally in the ratio $m:n$. The coordinates of such a point is given as $x=\dfrac{n{{x}_{1}}+m{{x}_{2}}}{m+n}$ and $y=\dfrac{n{{y}_{1}}+m{{y}_{2}}}{m+n}$. So you must remember the section formula because even if you forget the mid – point formula you may use these relations to derive them. Remember that in the case of mid – point we have $m:n=1:1$.