Problem 12 - Algebra 1: Concepts and Skills

Question

Find the midpoint of the line segment with the given endpoints. Then show that the midpoint is the same distance from each given point. \((-6,0),(-10,-2)\)

Step-by-Step Solution

Answer

The midpoint of the line segment having endpoints (-6,0) and (-10,-2) is (-8, -1), and this point is equidistant from the two given points, with the distance between the midpoint and each endpoint being \(\sqrt{5}\).

Step 1: Find the Midpoint

The midpoint formula is \((\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2})\), applied to our given data points \((-6,0)\) and \((-10,-2)\), the midpoint is calculated as \(\left(\frac{{-6 + -10}}{2}, \frac{{0 + -2}}{2}\right)\). This results in a midpoint of \((-8,-1)\).

Step 2: Calculate Distances

The distance formula is \(d = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}}\). To verify that the midpoint is equidistant from the two endpoints, calculate the distance between the midpoint \((-8,-1)\) and each of the given points. Distance between the midpoint and the point \((-6,0)\) is calculated as \(d1 = \sqrt{{(-8 - -6)^2 + (-1 - 0)^2}}\), yielding \(d1 = \sqrt{5}\). Distance between the midpoint and the point \((-10,-2)\) is calculated as \(d2 = \sqrt{{(-8 - -10)^2 + (-1 - -2)^2}}\), yielding \(d2 = \sqrt{5}\).

Step 3: Conclusion

As observed, both distances \(d1\) and \(d2\) are equal. Therefore, the midpoint \(M\), found to be \((-8,-1)\), is indeed equidistant from both given points \((-6,0)\) and \((-10,-2)\).

Key Concepts

Coordinate Geometry

In the fascinating world of mathematics, coordinate geometry, also known as analytic geometry, is a powerful branch that allows us to analyze geometric shapes and relationships through a system of coordinates. It is essentially a merge of algebra and geometry where points are placed on a grid called the coordinate plane. This grid is defined by a horizontal axis, known as the X-axis, and a vertical axis, dubbed the Y-axis.

Each point on this plane is represented by a pair of numerical values \( (x, y) \) that correspond to its horizontal and vertical positions, respectively. By using these coordinates, we can plot points, draw line segments, and figure out all sorts of calculations such as distances, midpoints, and more.

Distance Formula

The distance formula is a fundamental tool in coordinate geometry that enables us to calculate the precise distance between two points in a plane. It's rooted in the Pythagorean Theorem, a principle you might recall from studying right triangles.

Given two points, \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the distance between them is expressed as \( d = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}} \). This formula reveals the direct line distance 'as the crow flies' and not considering any physical barriers that may exist on a real-world map. It's a crucial concept that students need to grasp for a variety of applications within mathematics, physics, and beyond.

Midpoint of a Line Segment

Understanding the midpoint of a line segment is vital in dividing any given segment into two equal parts. The midpoint formula \( (\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}) \) takes the average of the x-coordinates and y-coordinates of the endpoints to find the center point. It's like finding the heart of the segment, equally accessible from both ends.

For instance, if we have endpoints \( A(-6, 0) \) and \( B(-10, -2) \) as in the given problem, we apply the formula to obtain the midpoint \( M(-8, -1) \). This point is not only central in its placement but holds the characteristic of being the same distance from each of the endpoints. The beauty of the midpoint is that it provides a simple yet profound way to understand symmetric relationships in geometry.