Question

How do you apply the midpoint formula when the coordinates are fractions?

Answer 1

Hint: We explain the internal ratio for given two points and find the formula. We then take the midpoint theorem where the internal ration is $1:1$. We also form the new formula for the midpoint case. We take two arbitrary fraction points and find the midpoint. We also use the real valued points to verify the result.

Complete step by step answer:
The general formula of ratio division of two points $\left( a,b \right)$ and $\left( c,d \right)$ with internal division ratio $m:n$ is $\left( \dfrac{mc+na}{m+n},\dfrac{md+nb}{m+n} \right)$.
In case of midpoint, the internal ratio becomes $1:1$.
The general formula for the midpoint of points $\left( a,b \right)$ and $\left( c,d \right)$ will be $\left( \dfrac{a+c}{2},\dfrac{b+d}{2} \right)$.
We have to find the case when the coordinates are fraction.
In those cases, we have to take the sum of the fractions and then divide by 2. In that case we just need to multiply 2 in the denominator.
Let’s take a general example and a real valued example.
Let two fractions be $\dfrac{p}{q}$ and $\dfrac{a}{b}$. The summation will be
$\dfrac{p}{q}+\dfrac{a}{b}=\dfrac{pb+aq}{bq}$.
Then we take the division by 2 which gives us $\dfrac{pb+aq}{2bq}$.
Now we take one real valued example.
We take two fractions $\dfrac{2}{3}$ and $\dfrac{3}{5}$.
The summation will be $\dfrac{2}{3}+\dfrac{3}{5}=\dfrac{10+9}{15}=\dfrac{19}{15}$.
Then we take the division by 2 which gives us $\dfrac{19}{2\times 15}=\dfrac{19}{30}$.

Note: The difference between using fractions and integer isn’t different. The process is similar for both cases. In case of integers, we don’t have to find the L.C.M of the denominators. The summation only works differently.