Question

The coordinates of the midpoint of the CD are (1, m) what is the value of m?

Answer 1

Hint: For the type of question that has been mentioned above we will first need to assume the x and y coordinates of the points that has been mentioned as C and D whose midpoint is given then we will use the basic formula of midpoint to find the value of m for which the question is about.

Complete step by step solution:
So in the above question it is mentioned that the midpoint of CD is (1, m) so now to find the value of m we are first going to assume the x and y coordinates of C and D to be \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] respectively. So we got coordinates of C to be \[\left( {{x}_{1}},{{y}_{1}} \right)\] and D to be \[\left( {{x}_{2}},{{y}_{2}} \right)\].

Now that we know the x and y coordinates of the two points for which we know the midpoint coordinates we will just use the midpoint formula which states that: \[x=\left( \dfrac{{{x}_{1}}+{{x}_{2}}}{2} \right)\] and \[y=\left( \dfrac{{{y}_{1}}+{{y}_{2}}}{2} \right)\] respectively, where x and y are the midpoint coordinates of CD. With this formula at hand we can easily substitute the values in the formula to get to the final answer. Now taking the midpoint coordinates mentioned in the question and also with the help of assumed coordinates of x and y we can easily use the midpoint formula for y to get to the final answer which will give us the final value of m, so now we will substitute y = m and for\[{{y}_{1}}\],\[{{y}_{2}}\] we will leave it as it is, as we have assumed the y coordinates to be \[{{y}_{1}}\],\[{{y}_{2}}\]now we will get the value of m as
\[\begin{align}
  & \Rightarrow m=\left( \dfrac{{{y}_{1}}+{{y}_{2}}}{2} \right) \\
 & \Rightarrow 2m={{y}_{1}}+{{y}_{2}} \\
\end{align}\]
From this we can finally say that the value of m is equal to \[\left( \dfrac{{{y}_{1}}+{{y}_{2}}}{2} \right)\]

Note: For the type of questions that we can see above, we can clearly see that the basic formulas are required to solve even the complicated question, in the question above we don’t know the coordinates of C and D but with the help of the formula we were still able to find out the value of m with the help of formula. So, try to remember these basic formulas which will be used later.