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

Hint:
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 \[\left( {\dfrac{{m{x_1} + n{x_2}}}{{m + n}},\dfrac{{m{y_1} + n{y_2}}}{{m + n}}} \right)\].