Question

The coordinates of the midpoint of a line segment $AB$ are $\left( {1, - 2} \right)$. If the coordinates of $A$ are $\left( { - 3,2} \right)$, then the coordinates of $B$ are
(a) $\left( {3, - 5} \right)$
(b) $\left( {5, - 6} \right)$
(c) $\left( {4, - 2} \right)$
(d) $\left( {5, - 4} \right)$

Answer 1

Here, we need to find the coordinates of $B$. Let the coordinates of the point $B$ be $\left( {a,b} \right)$. We will use the midpoint formula to form two linear equations in one variable. Then, we will solve these equations separately to get the coordinates of the point $B$.
Formula Used: According to the midpoint formula, the coordinates of the mid-point of the line segment joining two points $P\left( {{x_1},{y_1}} \right)$ and $Q\left( {{x_2},{y_2}} \right)$ are given by $\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)$.

Complete step by step solution:
Let the coordinates of the point $B$ be $\left( {a,b} \right)$.
According to the midpoint formula, the coordinates of the mid-point of the line segment joining two points $P\left( {{x_1},{y_1}} \right)$ and $Q\left( {{x_2},{y_2}} \right)$ are given by $\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)$.
The coordinates of the mid-point of the line segment joining the points A$$$$\left( { - 3,2} \right) and B$$$$\left( {a,b} \right) are $\left( {1, - 2} \right)$.
Therefore, substituting ${x_1} = - 3$, ${y_1} = 2$, ${x_2} = a$, and ${y_2} = b$ in the midpoint formula, we get
$\Rightarrow \left( {1, - 2} \right) = \left( {\dfrac{{ - 3 + a}}{2},\dfrac{{2 + b}}{2}} \right)$
Comparing the abscissa and the ordinate, we get the equations
$\Rightarrow 1 = \dfrac{{ - 3 + a}}{2}$ and $- 2 = \dfrac{{2 + b}}{2}$
We will solve these equations to get the values of $a$ and $b$, and hence, the coordinates of point $B$.
First, we will solve the equation $1 = \dfrac{{ - 3 + a}}{2}$.
Multiplying both sides of the equation $1 = \dfrac{{ - 3 + a}}{2}$ by 2, we get
\Rightarrow 1 \times 2 = \dfrac{{ - 3 + a}}{2} \times 2 \\\ \Rightarrow 2 = - 3 + a \\\
Adding 3 to both sides of the equation, we get
$\Rightarrow 2 + 3 = - 3 + a + 3$
Therefore, we get
$\therefore a = 5$
Now, we will solve the equation $- 2 = \dfrac{{2 + b}}{2}$.
Multiplying both sides of the equation $- 2 = \dfrac{{2 + b}}{2}$ by 2, we get
\Rightarrow - 2 \times 2 = \dfrac{{2 + b}}{2} \times 2 \\\ \Rightarrow - 4 = 2 + b \\\
Subtracting 2 from both sides of the equation, we get
$\Rightarrow - 4 - 2 = 2 + b - 2$
Therefore, we get
$\therefore b = - 6$
We get $\left( {a,b} \right) = \left( {5, - 6} \right)$.

Therefore, the coordinates of point $B$ are $\left( {5, - 6} \right)$. The correct option is option (b).

Note:
We used the terms ‘abscissa’ and ‘ordinate’ in the solution. The abscissa of a point $\left( {x,y} \right)$ is $x$, and the ordinate of a point $\left( {x,y} \right)$ is $y$.
The midpoint formula is derived from the section formula, where the ratio in which the line segment is divided is $1:1$. According to the section formula, the coordinates of a point dividing the line segment joining two points $P\left( {{x_1},{y_1}} \right)$ and $Q\left( {{x_2},{y_2}} \right)$ in the ratio $m:n$, are given by $\left( {\dfrac{{m{x_1} + n{x_2}}}{{m + n}},\dfrac{{m{y_1} + n{y_2}}}{{m + n}}} \right)$.