Solving for Point B Coordinates Using Midpoint Formula

The problem asks us to find the coordinates of point B, given the coordinates of point A (2, 4) and the midpoint D (-1, 3) of the line segment AB.

Applying the Midpoint Formula

Let the coordinates of A be \((x_1, y_1) = (2, 4)\).

Let the coordinates of B be \((x_2, y_2)\).

Let the coordinates of the midpoint D be \((x_m, y_m) = (-1, 3)\).

The midpoint formula states that:

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

Step-by-Step Calculation

1. Equate x-coordinates:

   Use the x-coordinate part of the midpoint formula:

   \(x_m = \frac{x_1 + x_2}{2}\)

   Substitute the known values:

   \(-1 = \frac{2 + x_2}{2}\)

   Multiply both sides by 2:

   \(-1 \times 2 = 2 + x_2\)

   \(-2 = 2 + x_2\)

   Solve for \(x_2\):

   \(x_2 = -2 - 2\)

   \(x_2 = -4\)

2. Equate y-coordinates:

   Use the y-coordinate part of the midpoint formula:

   \(y_m = \frac{y_1 + y_2}{2}\)

   Substitute the known values:

   \(3 = \frac{4 + y_2}{2}\)

   Multiply both sides by 2:

   \(3 \times 2 = 4 + y_2\)

   \(6 = 4 + y_2\)

   Solve for \(y_2\):

   \(y_2 = 6 - 4\)

   \(y_2 = 2\)

3. Conclusion:

   The calculated coordinates for point B are \((-4, 2)\).

Verifying the Answer

Comparing the calculated coordinates \((-4, 2)\) with the given options, Option 3 matches our result.