Chapter 1.8, Problem 46E

To determine

To calculate: The coordinates of B.

Answer to Problem 46E

The coordinates of the point B is $\left(10,13\right)$ .

Explanation of Solution

Given information:

  $M\left(6,8\right)$ is the midpoint of the line segment $AB$ and A has coordinates $\left(2,3\right)$ .

Formula used:

Mid-point formula between two points $\text{X}\left(a_1,b_1\right)$ and $\text{Y}\left(a_2,b_2\right)$ is mathematically expressed as,

  $\left(\frac{a_1+a_2}{2},\frac{b_1+b_2}{2}\right)$

Calculation:

Consider the given information that $M\left(6,8\right)$ is the midpoint of the line segment $AB$ and A has coordinates $\left(2,3\right)$ .

Let us assume coordinates of point B as $\left(x,y\right)$ .

Recall the mid-point formula between two points $\text{X}\left(a_1,b_1\right)$ and $\text{Y}\left(a_2,b_2\right)$ is mathematically expressed as,

  $\left(\frac{a_1+a_2}{2},\frac{b_1+b_2}{2}\right)$

So, mid-point of line segment $AB$ is calculated as,

  $\left(\frac{2+x}{2},\frac{3+y}{2}\right)$

But it is given that mid-point of $AB$ is $\left(6,8\right)$ ,

So, it will be further solved by equating the both,

  $\left(\frac{2+x}{2},\frac{3+y}{2}\right)=\left(6,8\right)$

Equate x-coordinates and y-coordinates on both sides,

  $\begin{array}{c}\frac{2+x}{2}=6\\ 2+x=12\\ x=12-2\\ x=10\end{array}$

  $\begin{array}{c}\frac{3+y}{2}=8\\ 3+y=16\\ y=16-3\\ y=13\end{array}$

Thus, the coordinates of the point B is $\left(10,13\right)$ .