Problem 10 - Algebra 2

Question

Find the midpoint of the line segment with endpoints at the given coordinates. $$ (8,3),(16,7) $$

Step-by-Step Solution

Answer

The midpoint is (12, 5).

Step 1: Identify Endpoint Coordinates

We have two endpoints of a line segment: \((x_1, y_1) = (8, 3)\) and \((x_2, y_2) = (16, 7)\). These coordinates will be used to find the midpoint.

Step 2: Apply the Midpoint Formula

The formula for finding the midpoint \((x_m, y_m)\) of a line segment is given by:\[ x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2} \]We will substitute the values of \(x_1, y_1, x_2,\) and \(y_2\) into this formula.

Step 3: Calculate the Midpoint for x-coordinate

Substitute \(x_1 = 8\) and \(x_2 = 16\) into the formula: \[ x_m = \frac{8 + 16}{2} = \frac{24}{2} = 12 \]This gives us the x-coordinate of the midpoint as 12.

Step 4: Calculate the Midpoint for y-coordinate

Substitute \(y_1 = 3\) and \(y_2 = 7\) into the formula: \[ y_m = \frac{3 + 7}{2} = \frac{10}{2} = 5 \]This gives us the y-coordinate of the midpoint as 5.

Step 5: Combine the Midpoint Coordinates

Combine the calculated x and y coordinates to get the midpoint. The midpoint of the line segment is \((12, 5)\).

Key Concepts

Understanding Coordinates

Coordinates are fundamental in both algebra and geometry. They help us pinpoint exact locations on a plane. In a coordinate system, each point is represented by a pair of numbers: \(x, y\). This pair denotes the horizontal (x) and vertical (y) positions respectively. For example:

• The point \(8, 3\) has an x-coordinate of 8 and a y-coordinate of 3.
• The point \(16, 7\) has an x-coordinate of 16 and a y-coordinate of 7.

These coordinates help us understand both the location and the relation between points, which is essential when solving problems like finding midpoints, distances or understanding line segments.

Defining a Line Segment

A line segment is a part of a line that has two endpoints. Unlike a line, it does not extend infinitely. When you have two points like \(8, 3\) and \(16, 7\), you can connect them with a straight path on a plane.
This path is known as a line segment. Think of it like a piece of string stretching between two nails at specified coordinates. Line segments are foundational in geometry, helping us understand shapes, angles, and distances.
They are used to construct polygons, determine angles, and calculate lengths among other applications.

Applying Algebra in Geometry

Algebra and geometry might seem like distinct areas of mathematics, but they often overlap. In this exercise, we use algebraic formulas to solve geometric problems. Here’s how algebra helps:

• Formulas: Algebra provides formulas, like the midpoint formula, to find specific points between two coordinates.
• Calculations: It allows us to compute precise figures using operations like addition and division.
• Coordination between numbers: It helps in organizing data systematically, ensuring that steps lead to accurate solutions.

For example, by substituting \(x_1 = 8\) and \(x_2 = 16\) into \[(x_m = \frac{x_1 + x_2}{2})\], algebra allows us to calculate the exact midpoint with ease.

Exploring Geometry in Practice

Geometry is the study of shapes, sizes, and the properties of space. It provides tools like the midpoint formula which simplify solving geometric queries. In practical terms:

• Visual Understanding: Geometry transforms abstract concepts into visible and measurable figures.
• Real-world Applications: Concepts like line segments and midpoints are applied in fields ranging from art to engineering.
• Spatial Reasoning: Understanding geometry fosters strong analytical and reasoning skills, as it involves visualizing shapes and structures.

By calculating the midpoint \(12, 5\) for the line segment joining coordinates \(8, 3\) and \(16, 7\), students get a concrete handle on abstract calculations, enriching their comprehension of the spatial world.