$M$ is the midpoint of $\overset{―}{A N}, A$ has coordinates $(2, - 3)$ , and $M$ has coordinates $(5, 1)$ . Find the coordinates of $N$ .
$(8, 5)$
$(7, - 2)$
$(9 \frac{1}{2}, 4)$
$(3 \frac{1}{2}, - 1)$

Answer

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Answer

Therefore, the coordinates of point $N$ are $(8, 5)$ .

Steps

Step 1 ：Use the midpoint formula: $M = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$

Step 2 ：Substitute the given values: $M = (\frac{2 + x_{2}}{2}, \frac{- 3 + y_{2}}{2})$

Step 3 ：Simplify the equations: $5 = \frac{2 + x_{2}}{2}$ and $1 = \frac{- 3 + y_{2}}{2}$

Step 4 ：Multiply both sides of the first equation by 2: $10 = 2 + x_{2}$

Step 5 ：Subtract 2 from both sides: $x_{2} = 8$

Step 6 ：Multiply both sides of the second equation by 2: $2 = - 3 + y_{2}$

Step 7 ：Add 3 to both sides: $y_{2} = 5$

Step 8 ：Therefore, the coordinates of point $N$ are $(8, 5)$ .