Question

If the midpoint of the segment AB is (1, 4), the coordinates of A are (2, – 3). How do you find the coordinate of point B?

Answer 1

Hint: We are given that the midpoint of segment AB is (1, 4) and we are given the coordinate of one endpoint and we are asked to find the other. To find this we will first learn how the points and the ratio in which the points divide the line segmented connected are. Then we will use the section formula \[X=\dfrac{{{m}_{1}}{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}}\] and \[Y=\dfrac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}}.\] We will use the coordinate of A and the midpoint to find the coordinate of B. We will also learn about a short trick to do such problems.

Complete step by step answer:
We are given that we have a line segment AB whose midpoint is (1, 4). The midpoint is the points that will divide the line into two equal ratios. If we consider that X is the midpoint of AB then AX = XB. So, the ratio of AX:XB will be

\[\dfrac{AX}{XB}=\dfrac{AX}{AX}\left[ \text{As }XB=AX \right]\]
\[\Rightarrow \dfrac{AX}{XB}=\dfrac{1}{1}\]
Therefore the ratio is 1:1.
Now, we will use the section formula. The section formula tells us about how the points cutting the line segment into sections are connected to the coordinate of the endpoint of the line segment. If we have

X divides it into ratio \[{{m}_{1}}:{{m}_{2}}\] then the coordinate of X is given as
\[X=\dfrac{{{m}_{1}}{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}}\]
\[Y=\dfrac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}}\]
In our problem we have coordinates of A as (2, – 3). So, \[\left( {{x}_{1}},{{y}_{1}} \right)=\left( 2,-3 \right)\] and the coordinate of midpoint X as (1, 4). So, (x, y) = (1, 4). We have to find the value of coordinates of B, we have it as \[B\left( {{x}_{2}},{{y}_{2}} \right).\] So as we have that ratio as 1:1, so \[{{m}_{1}}=1\] and \[{{m}_{2}}=1.\] Now, we use these values in
\[X=\dfrac{{{m}_{1}}{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}}\]
\[\Rightarrow 1=\dfrac{1\times {{x}_{2}}+1\left( 2 \right)}{1+1}\]
On simplifying, we get,
\[\Rightarrow 1=\dfrac{{{x}_{2}}+2}{2}\]
Solving for \[{{x}_{2}}\] we get,
\[\Rightarrow {{x}_{2}}+2=2\]
\[\Rightarrow {{x}_{2}}=0\]
Now, using the value on \[Y=\dfrac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}}\] we get
\[\Rightarrow 4=\dfrac{1\times {{y}_{2}}+\left( -3 \right)}{1+1}\]
On simplifying, we get,
\[\Rightarrow 4=\dfrac{{{y}_{2}}-3}{2}\]
Solving for \[{{y}_{2}}\] we get,
\[\Rightarrow 8={{y}_{2}}-3\]
\[\Rightarrow {{y}_{2}}=11\]
So, we get \[{{x}_{2}}=0\] and \[{{y}_{2}}=11.\] So, the coordinate of B is (0, 11)

Note: We can also solve the problem in which are given the midpoint, as we have a midpoint formula given to us as \[x=\dfrac{{{x}_{1}}+{{x}_{2}}}{2}\] and \[y=\dfrac{{{y}_{1}}+{{y}_{2}}}{2}.\] Now as x = 1 and \[{{x}_{1}}=2,\] so we get,
\[\Rightarrow 1=\dfrac{2+{{x}_{2}}}{2}\]
On simplifying, we get,
\[\Rightarrow 2=2+{{x}_{2}}\]
Hence,
\[\Rightarrow {{x}_{2}}=0\]
Now, we have \[{{y}_{1}}=-3\] and y = 4, so we get,
\[4=\dfrac{-3+{{y}_{2}}}{2}\]
On simplifying, we get,
\[8=-3+{{y}_{2}}\]
Solving for \[{{y}_{2}}\] we get,
\[\Rightarrow {{y}_{2}}=11\]
So, the coordinate of B is \[\left( {{x}_{2}},{{y}_{2}} \right)=\left( 0,11 \right).\]