Problem 29 - Precalculus Essentials

Question

Find the midpoint of each line segment with the given endpoints. $$(\sqrt{18},-4) \text { and }(\sqrt{2}, 4)$$

Step-by-Step Solution

Answer

\((M_x, M_y) = \left(\frac{\sqrt{18}+\sqrt{2}}{2}, 0\right)\

Step 1: Identify the endpoints

The endpoints of the line segment are given as \((\sqrt{18}, -4)\) and \((\sqrt{2}, 4)\). So, \(x_1=\sqrt{18}\), \(y_1=-4\), \(x_2=\sqrt{2}\), and \(y_2=4\).

Step 2: Apply the midpoint formula

Substitute \(x_1\), \(y_1\), \(x_2\), and \(y_2\) into the midpoint formula to find the midpoint. The midpoint \((M_x, M_y)\) is given by \((M_x, M_y) = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\)

Step 3: Calculate the midpoint

Substitute the values into the formula to calculate the midpoint: \((M_x, M_y) = \left(\frac{\sqrt{18}+\sqrt{2}}{2}, \frac{-4+4}{2}\right)\)

Step 4: Simplify the result

Simplify the result to find the midpoint: \((M_x, M_y) = \left(\frac{\sqrt{18}+\sqrt{2}}{2}, 0\right)\)

Key Concepts

Coordinate Geometry

Coordinate geometry, often known as analytic geometry, provides us with the tools to describe and manipulate the positions of points on a plane using algebraic equations. This branch of mathematics is vital because it allows us to solve complex problems related to points, lines, and shapes with the help of coordinates and formulas.
For instance, the midpoint formula is an essential tool in coordinate geometry. It calculates the exact middle point of a line segment given the coordinates of its endpoints.

• The formula for the midpoint \(M(x, y)\) between two endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \((M_x, M_y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\).
• The x-coordinate of the midpoint is the average of the x-coordinates of the endpoints.
• The y-coordinate of the midpoint is the average of the y-coordinates of the endpoints.

Understanding how to use this formula enables us to determine the exact location of a midpoint in a coordinate plane, which is crucial in many geometry problems.

Radical Expressions

Radical expressions, commonly featuring square roots and other roots, appear frequently in coordinate geometry when dealing with non-integer coordinates. Knowing how to manipulate these expressions is essential for simplifying and solving equations.
In our exercise, we encounter radical expressions because the coordinates involve square roots: \(\sqrt{18}\) and \(\sqrt{2}\). To handle these, you can use properties of radicals to simplify and better manage calculations.

• Combine like terms: Add and subtract expressions in the same root form.
• Simplifying radicals: Identify squares within the radicand (the number under the root sign) to simplify expressions, such as \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\).
• Exact versus decimal forms: While it's possible to format solutions decimally, using exact forms with radicals keeps calculations precise.

Handling radicals accurately ensures you can solve geometry problems that involve irrational numbers, maintaining precision and understanding.

Line Segments

A line segment in geometry is part of a line defined by two endpoints. These endpoints give the segment a definite length and determine its position in space. Connecting these points creates a finite part of the line, known simply as the line segment.
Understanding line segments is crucial when using the midpoint formula. The formula is applied to line segments because they have distinct endpoints, unlike lines that extend indefinitely in both directions.

• Endpoints in the exercise: \((\sqrt{18}, -4)\) and \((\sqrt{2}, 4)\). These points anchor the segment.
• Midpoint significance: The midpoint of a line segment like \(\left(\frac{\sqrt{18}+\sqrt{2}}{2}, 0\right)\) divides the segment into two equal parts, revealing symmetry and balance.
• Application: Identifying the midpoint of line segments helps in geometric constructions and is used in various real-world applications such as computer graphics and architecture.

Grasping how to work with line segments, understand their nature and use their coordinate points enhances problem-solving skills in geometry and beyond.