Find the midpoint of each line segment with the given endpoints. (7√3, −6) and (3√3, −2)

Verified step by step guidance

1. Recall the midpoint formula for a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]
2. Identify the coordinates of the given endpoints: \(x_1 = 7\sqrt{3}\), \(y_1 = -6\), \(x_2 = 3\sqrt{3}\), and \(y_2 = -2\).
3. Substitute the \(x\)-coordinates into the midpoint formula: \[\frac{7\sqrt{3} + 3\sqrt{3}}{2}\]
4. Substitute the \(y\)-coordinates into the midpoint formula: \[\frac{-6 + (-2)}{2}\]
5. Simplify both expressions to find the midpoint coordinates, which will be the average of the \(x\)-values and the average of the \(y\)-values.

Key Concepts

Midpoint Formula

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

Coordinate Geometry

Coordinate geometry involves representing geometric figures using coordinates on the Cartesian plane. Understanding how points, lines, and shapes are expressed with ordered pairs (x, y) 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 answers are presented clearly and correctly, such as simplifying expressions like 7√3 or 3√3.