We are given the following information:

• The midpoint $M(1, 3)$
• Point $A(3, 1)$
• We need to find the coordinates of point $B$.

The formula for the midpoint between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Substitute the known values of $M$ and $A$:

$(1, 3) = \left( \frac{3 + x_2}{2}, \frac{1 + y_2}{2} \right)$

This gives two equations:

1. $\frac{3 + x_2}{2} = 1$
2. $\frac{1 + y_2}{2} = 3$

Solve each equation:

For $x_2$:

$\frac{3 + x_2}{2} = 1$ Multiply both sides by 2: $3 + x_2 = 2$ Subtract 3 from both sides: $x_2 = -1$

For $y_2$:

$\frac{1 + y_2}{2} = 3$ Multiply both sides by 2: $1 + y_2 = 6$ Subtract 1 from both sides: $y_2 = 5$

Thus, the coordinates of point $B$ are $(-1, 5)$.

Would you like further details or have any questions? Here are some related questions:

1. How can the midpoint formula be derived from geometry?
2. What is the distance between points $A$ and $B$?
3. How would the solution change if the midpoint was given in 3D space?
4. What are some real-life applications of midpoint calculations?
5. How can the midpoint be used to divide a line segment in a different ratio?

Tip: When solving for an unknown in midpoint problems, always work with each coordinate (x and y) separately.