Question

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

Answer 1

Hint: Assume the point whose coordinates we need to find as $B(x_2,y_2)$, coordinates of A as $A(x_1,y_1)$ and the mid – point M as $M(x,y)$. Now, apply the mid – point formula for the coordinates x and y given as $x={\displaystyle \frac{x_1+x_2}{2}}$ and $y={\displaystyle \frac{y_1+y_2}{2}}$. Substitute all the given values and determine the coordinates of $(x_2,y_2)$ 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 $A(x_1,y_1)$, the mid – point M as $M(x,y)$ and the missing point B as $B(x_2,y_2)$ 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 $x={\displaystyle \frac{x_1+x_2}{2}}$ and the y – coordinate as $y={\displaystyle \frac{y_1+y_2}{2}}$. Let us solve for them one by one.
(i) For x – coordinate we have,
$\Rightarrow x={\displaystyle \frac{x_1+x_2}{2}}$
Substituting the given values in the above relation we get,
$\Rightarrow 5={\displaystyle \frac{2+x_2}{2}}$
By cross – multiplication we get,
$\begin{array}{rl}& \Rightarrow 10=2+x_2\\ & \Rightarrow 10-2=x_2\\ & \therefore x_2=8\end{array}$
(i) For y – coordinate we have,
$y={\displaystyle \frac{y_1+y_2}{2}}$
Substituting the given values in the above relation we get,
$\Rightarrow 4={\displaystyle \frac{2+y_2}{2}}$
By cross – multiplication we get,
$\begin{array}{rl}& \Rightarrow 8=2+y_2\\ & \Rightarrow 8-2=y_2\\ & \therefore y_2=6\end{array}$
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 $$x={\displaystyle \frac{n{x}_1+m{x}_2}{m+n}}$$ and $$y={\displaystyle \frac{n{y}_1+m{y}_2}{m+n}}$$. 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 $m:n=1:1$.