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

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

& \Rightarrow x+5-5=12-5 \\\ & \therefore x=7 \\\ \end{aligned}$$ Similarly, $$\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{aligned} & \Rightarrow y+3-3=8-3 \\\ & \therefore y=5 \\\ \end{aligned}$$ Thus, we get the coordinates of the other points as $$\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.