Problem 56 - Algebra 2

Question

Find the midpoint of the line segment with endpoints having the given coordinates. $$ (8,0),(-5,12) $$

Step-by-Step Solution

Answer

The midpoint is (1.5, 6).

Step 1: Understand the Midpoint Formula

The midpoint of a line segment with endpoints \(x_1, y_1\) and \(x_2, y_2\) is calculated using the formula: \[(x_m, y_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]This formula gives the average of the x-coordinates and the y-coordinates individually to find the center point of the segment.

Step 2: Substitute the Endpoint Coordinates

Substitute the given coordinates \(x_1 = 8, y_1 = 0\) and \(x_2 = -5, y_2 = 12\) into the midpoint formula. It becomes:\[(x_m, y_m) = \left( \frac{8 + (-5)}{2}, \frac{0 + 12}{2} \right)\]

Step 3: Calculate the Average of the x-coordinates

Calculate the x-coordinate of the midpoint by taking the average of 8 and -5:\[\frac{8 + (-5)}{2} = \frac{3}{2} = 1.5\]

Step 4: Calculate the Average of the y-coordinates

Calculate the y-coordinate of the midpoint by averaging 0 and 12:\[\frac{0 + 12}{2} = \frac{12}{2} = 6\]

Step 5: Formulate the Midpoint

Combine the calculated x- and y-coordinates to write the coordinates of the midpoint:The midpoint is \(1.5, 6\).

Key Concepts

The Anatomy of a Line Segment

A line segment is a fundamental concept in geometry. It is a part of a line that is bounded by two distinct endpoints. Unlike a line, which extends infinitely in both directions, a line segment has a clear beginning and end, making it measurable. This measurability is crucial in problems involving the midpoint formula.

To visualize a line segment, imagine it as a straight piece of string. The endpoints denote where the string starts and stops. When we measure or perform calculations involving a line segment, we focus exclusively on the distance and position between these two endpoints.

Key Points about Line Segments:

• A line segment is defined by its two endpoints.
• It is finite and has a specific length.
• End points are crucial for calculating the midpoint.

What is Coordinate Geometry?

Coordinate geometry, often referred to as analytic geometry, is a branch of mathematics that uses algebraic techniques to solve geometric problems. It involves placing geometric figures into a coordinate system, typically on a 2D plane like the Cartesian plane, which includes a horizontal x-axis and a vertical y-axis.

The connection between algebra and geometry through coordinate systems allows us to solve complex geometric problems with more straightforward algebraic methods. In the context of finding a midpoint, coordinate geometry helps visualize the problem by plotting points and calculating based on their positions.

Essential Elements of Coordinate Geometry:

• Points have coordinates, typically written as (x, y).
• It allows complex geometric problems to be solved algebraically.
• Helps in understanding the position and distance between points.

Understanding Averages and Their Role

The concept of an average is at the heart of many mathematical problems, including those involving the midpoint formula. An average is a single value that represents the central tendency of a set of numbers. In the case of the midpoint of a line segment, an average is used to find the point that lies exactly halfway between two endpoints.

When dealing with coordinates, the average is computed by adding two numbers together and dividing by two. This calculation ensures each coordinate shifts equally towards the center, giving the midpoint.

Why Averages Matter in Midpoint Calculation:

• An average finds the central point between two numbers.
• It ensures even spacing from both endpoints to the midpoint.
• Midpoint uses the average to provide equal representation for x and y coordinates.