If point A is located at \((3, 4)\) and point B is at \((-2, -1)\), what are the coordinates of the midpoint between points A and B?

To find the midpoint between two points, you can use the midpoint formula: \[ \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \] For points A \((3, 4)\) and B \((-2, -1)\): - \(x_1 = 3\), \(y_1 = 4\) - \(x_2 = -2\), \(y_2 = -1\) Plugging these values into the formula: \[ \text{Midpoint} = \left(\frac{3 + (-2)}{2}, \frac{4 + (-1)}{2}\right) = \left(\frac{3 - 2}{2}, \frac{4 - 1}{2}\right) = \left(\frac{1}{2}, \frac{3}{2}\right) \] So, the coordinates of the midpoint between points A and B are \(\left(\frac{1}{2}, \frac{3}{2}\right)\). Now, let’s dive into some interesting additional knowledge! Midpoints have a rich history dating back to ancient mathematics, where Greco-Roman scholars calculated them for geometric purposes. They played a critical role in constructing bisectors and understanding symmetry in shapes, laying foundational work for modern geometry. In real-world applications, midpoints are invaluable in fields like engineering and design, where they help in creating balanced structures. For instance, when designing a bridge, knowing the midpoint can assist in evenly distributing weight, ensuring stability, and enhancing safety for everyone crossing.