Find the midpoint of each line segment with the given endpoints. (8, 3√5) and (−6, 7√5)

Verified step by step guidance

1. Recall that the midpoint \( M \) of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ M = \left( \frac{ x_1 + x_2 }{ 2 }, \frac{ y_1 + y_2 }{ 2 } \right) \]
2. Identify the coordinates of the given endpoints: \( (x_1, y_1) = (8, 3\sqrt{5}) \) and \( (x_2, y_2) = (-6, 7\sqrt{5}) \).
3. Calculate the midpoint's \( x \)-coordinate by adding the \( x \)-values and dividing by 2: \[ \frac{8 + (-6)}{2} \]
4. Calculate the midpoint's \( y \)-coordinate by adding the \( y \)-values and dividing by 2: \[ \frac{3\sqrt{5} + 7\sqrt{5}}{2} \]
5. Combine the results to write the midpoint as: \[ \left( \frac{8 + (-6)}{2}, \frac{3\sqrt{5} + 7\sqrt{5}}{2} \right) \]

Key Concepts

Midpoint Formula

The midpoint formula calculates the point exactly halfway between two given points in a coordinate plane. It is found by averaging the x-coordinates and the y-coordinates separately: Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2). This formula helps locate the center of a line segment.

Coordinate Geometry

Coordinate geometry involves representing geometric figures using coordinates on the Cartesian plane. Understanding how points, lines, and shapes relate through their coordinates is essential for applying formulas like the midpoint formula and interpreting results accurately.

Simplifying Radicals

Simplifying radicals means expressing square roots in their simplest form by factoring out perfect squares. This skill is important when working with coordinates involving square roots, ensuring the final answer is presented clearly and correctly.