Question

How do you find the coordinates of the other endpoint of a segment with the given \[H(5,3)\] and the midpoint \[M(6,4)\]?

Answer 1

Hint: The midpoint of a segment is the point that divides the segment in half. We can find the mid-points of a segment, from its endpoints. Suppose we are given two points A and B, their coordinates are \[(a,b)\And (c,d)\] respectively. Then, the mid-point of the segment joining the points A and B has coordinates \[\left( \dfrac{a+c}{2},\dfrac{b+d}{2} \right)\].

Complete step-by-step answer:
We are given two points \[H(5,3)\], and \[M(6,4)\]. We know that M is the midpoint of the point H and one other point. Let the other point be \[G(x,y)\]. Using the mid-point theorem, we can say that
\[(6,4)=\left( \dfrac{x+5}{2},\dfrac{y+3}{2} \right)\]
Comparing the X and Y coordinate separately, we get can find the coordinates of G
\[\Rightarrow \dfrac{x+5}{2}=6\]
Multiplying both sides by 2, we get
\[\Rightarrow x+5=12\]
Subtracting 5 from both sides of the above equation, we get
\[\begin{align}
  & \Rightarrow x+5-5=12-5 \\
 & \therefore x=7 \\
\end{align}\]
Similarly,
\[\Rightarrow \dfrac{y+3}{2}=4\]
 Multiplying both sides by 2, we get
\[\Rightarrow y+3=8\]
Subtracting 3 from both sides of the above equation, we get
\[\begin{align}
  & \Rightarrow y+3-3=8-3 \\
 & \therefore y=5 \\
\end{align}\]
Thus, we get the coordinates of the other points as \[\left( 7,5 \right)\].

Note: We can use this question to make a general formula/ property to solve similar types of questions. Let’s say we are given two points having coordinates \[(a,b)\And (c,d)\] respectively. And we are told that the second point is the midpoint of the first and the other point. Then, we can find the coordinates of the other point as \[\left( 2c-a,2d-b \right)\]. By substituting the values of the coordinates of the two given points.