Problem 16 - Intermediate Algebra

Question

Verify that \((3,1)\) is the midpoint of the line segment joining \((-2,6)\) and \((8,-4)\).

Step-by-Step Solution

Answer

Yes, (3, 1) is the midpoint of the segment.

Step 1: Identify the Formula for Midpoint

The midpoint of a line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula: \( \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) \).

Step 2: Plug in the Given Points

Use the given points \((-2,6)\) as \((x_1, y_1)\) and \((8,-4)\) as \((x_2, y_2)\) in the formula. Plug these values into the formula: \( \left( \frac{-2+8}{2}, \frac{6+(-4)}{2} \right) \).

Step 3: Calculate the x-coordinate of the Midpoint

Calculate the x-coordinate of the midpoint: \( \frac{-2 + 8}{2} = \frac{6}{2} = 3 \).

Step 4: Calculate the y-coordinate of the Midpoint

Calculate the y-coordinate of the midpoint: \( \frac{6 + (-4)}{2} = \frac{2}{2} = 1 \).

Step 5: Write the Midpoint Coordinates

Combine the calculated coordinates to write the midpoint. The midpoint of the line segment is \((3,1)\).

Step 6: Verify with Given Point

The coordinates calculated for the midpoint \((3, 1)\) match the point given in the exercise. Hence, the verification is completed.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges the gap between algebra and geometry by allowing us to describe geometric figures using algebraic equations. This powerful branch of mathematics lets us study the properties and relationships of geometric shapes and their positions. By representing points on a plane as coordinate pairs, it becomes much simpler to calculate distances, slopes, and midpoints.

The coordinate plane consists of two intersecting number lines: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane is defined by a pair of numbers known as coordinates \(x, y\). These coordinates help us pinpoint the precise location of points, making complex geometric computations possible with ease.

Line Segment

A line segment is a part of a line that is bounded by two distinct endpoints, unlike a line which extends infinitely in both directions. Formally, a line segment joining two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) includes all the points on the line between \(A\) and \(B\).

In the context of coordinate geometry, line segments are crucial for calculating midpoints and distances, as they have both defined start and end points. While solving problems, knowing the endpoints of a line segment allows us to compute various properties such as length and the midpoint, providing deeper insights into the geometric properties of figures.

Mathematics Education

Mathematics education focuses on equipping students with a clear understanding of fundamental mathematical concepts and problem-solving skills. The study of coordinate geometry and line segments helps students not only work with geometric figures and their properties but also refine algebraic skills.

This fusion of algebra and geometry demonstrates how abstract mathematical ideas can be visually represented and manipulated. Resources such as step-by-step solutions and practical exercises make these concepts more approachable. By applying methods like the midpoint formula, students enhance their analytic skills, learning to verify and visualize geometric relationships effectively.

Problem Solving Steps

Problem-solving steps are essential for systematically tackling mathematical questions and proving solutions. For instance, verifying that a given point is the midpoint of a line segment involves a clear sequence of actions:

• First, identify the correct formula to use \(\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)\).
• Next, assign the given points to variables \(x_1, y_1, x_2, y_2\).
• Then, substitute these variables into the formula.
• Proceed with calculating the x and y coordinates of the midpoint separately.
• Finally, combine these coordinates to express the midpoint in its coordinate pair form and verify by matching it against the expected result.

The attention to detail in these steps ensures accuracy and fosters a deeper understanding, aiding students in honing their analytical and logical reasoning skills indispensable in mathematics.