Problem 30 - Precalculus Essentials

Question

Find the midpoint of each line segment with the given endpoints. $$(\sqrt{50},-6) \text { and }(\sqrt{2}, 6)$$

Step-by-Step Solution

Answer

The coordinates for the midpoint of this line segment are \((\frac{\sqrt{50} + \sqrt{2}}{2}, 0)\).

Step 1: Identify the coordinates of the endpoints

The coordinates of the first point are \((\sqrt{50}, -6)\) and of the second point are \((\sqrt{2}, 6)\). So \(X_1 = \sqrt{50}\), \(Y_1 = -6\), \(X_2 = \sqrt{2}\), and \(Y_2 = 6\).

Step 2: Apply the midpoint formula

Plug the values of \(X_1, X_2, Y_1, Y_2\) into the formula \((X_{mid} = (X_1 + X_2)/2, Y_{mid} = (Y_1 + Y_2)/2)\). This will give us our x and y coordinates of the midpoint.

Step 3: Compute the coordinates

Calculate the midpoint coordinates by computing the individual terms: \(X_{mid} = (\sqrt{50} + \sqrt{2})/2\) and \(Y_{mid} = (-6 + 6)/2\). Simplify those to get the final coordinates (X_{mid}, Y_{mid}).

Key Concepts

Distance Formula

The distance formula in mathematics helps us figure out how far apart two points are on a coordinate plane. This is especially useful in coordinate geometry, where we need to analyze line segments. The formula uses the Pythagorean Theorem and can be represented as follows:\[ d = \sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2} \]Here:

• \(X_1, Y_1\) are the coordinates of the first point
• \(X_2, Y_2\) are the coordinates of the second point
• \(d\) is the distance between the two points.

To apply this, you subtract the x-coordinates and y-coordinates of the points, square these differences, add them, and then take the square root of the result. This formula is essential in uncovering distances in all kinds of geometric contexts.

Line Segments

A line segment is a part of a line that is bounded by two distinct endpoints. In geometry, these segments are a fundamental concept because they help us understand more complex shapes and structures.

Key Characteristics of Line Segments

• Line segments have a clear starting and ending point.
• They are the shortest path connecting two points on a plane or in space.
• They do not extend infinitely like lines do.

When working with line segments, we often need to calculate properties like length using the distance formula, or the midpoint, which is the central point along the segment.

Coordinate Geometry

Coordinate geometry, also known as analytical geometry, is the study of geometry using a coordinate system. In this branch, we use algebraic methods to solve geometric problems. This approach allows us to describe the position and properties of objects in a plane, leveraging tools like the distance and midpoint formulas.

Importance of Coordinate Geometry

• Allows for precise descriptions of geometric figures.
• Facilitates calculations involving lines, angles, and distances.
• Connects algebraic concepts with geometric visuals.

With coordinate geometry, students can easily calculate distances between points, angles between lines, and other geometric properties. It provides a robust framework for exploring both simple and complex geometric problems.