Problem 8 - Algebra 1: Concepts and Skills

Question

Find the midpoint of the line segment with the given endpoints. \((2,1),(14,6)\)

Step-by-Step Solution

Answer

The midpoint of the line segment with endpoints \((2,1)\) and \((14,6)\) is \((8, 3.5)\).

Step 1: Identify the coordinates of the endpoints

The first endpoint's coordinates are \((2,1)\) and the second endpoint's are \((14,6)\). This corresponds with \((x_1, y_1)\) and \((x_2, y_2)\) in the midpoint formula.

Step 2: Apply the midpoint formula

Substitute the coordinates into the formula to find the midpoint: \(\left(\frac{2 + 14}{2}, \frac{1 + 6}{2}\right)\). After the calculation, it gives us \(\left(\frac{16}{2}, \frac{7}{2}\right)\)

Step 3: Simplify the expression

Simplify the expression to lowest form: The result is \((8, 3.5)\), which is the coordinates of the midpoint.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves the study of geometric figures through the use of a coordinate system. This allows for the manipulation of geometric problems into algebraic ones, making them easier to solve. In coordinate geometry, points on a plane are described by pairs of numerical coordinates, which are the distances from the point to two fixed perpendicular lines known as axes.

With coordinate geometry, you can calculate distances, midpoints, slopes, and more. In the context of the original exercise, it allows for a clear definition of a line segment by its endpoints and the calculation of its midpoint, giving students a visual understanding of what they are computing.

Finding Midpoints

The process of finding the midpoint of a line segment is crucial in both geometry and various real-world applications. The midpoint of a line segment is the point that divides the segment into two equal parts. It's like the 'halfway' point between the two ends. To find the midpoint, we use the midpoint formula which is derived from the averages of the x-coordinates and the y-coordinates of the endpoints.

The formula to find the midpoint \(M\) between two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is given by \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). When you replace \(x_1\), \(y_1\), \(x_2\), and \(y_2\) with their actual values, you simply compute the arithmetic mean of the x-coordinates and y-coordinates separately, resulting in the coordinates of the midpoint.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operators. They are fundamental in representing various mathematical ideas and solving equations and problems. When working with the midpoint formula, for example, we use algebraic expressions to represent the coordinates of points.

By substituting specific values into an algebraic expression, we can solve for unknowns or, as with the midpoint formula, determine new values based on existing ones. In the given exercise, the algebraic expressions involved are \(\frac{2 + 14}{2}\) and \(\frac{1 + 6}{2}\), which simplify to the numerical coordinates of the midpoint. Understanding how to manipulate and simplify these expressions is critical to solving problems in coordinate geometry.