Problem 17 - Algebra 1: Concepts and Skills

Question

Find the midpoint of the line segment connecting the given points. \((-1,2),(7,4)\)

Step-by-Step Solution

Answer

The midpoint of the line segment connecting the points (-1,2) and (7,4) is (3,3).

Step 1: Identify the coordinates of the two points

The coordinates of the first point, denoted as (x1, y1), are (-1,2). The coordinates of the second point, denoted as (x2, y2), are (7,4).

Step 2: Apply the midpoint formula

Substitute the coordinates of the two points into the midpoint formula, resulting in \((\frac{-1 + 7}{2}, \frac{2 + 4}{2})\)

Step 3: Simplify the expressions

Simplify the expressions in each coordinate to get the midpoint. This gives us \((\frac{6}{2}, \frac{6}{2})\) which simplifies to (3,3).

Key Concepts

Understanding Coordinates

Coordinates are a set of numbers that help us identify the exact location of a point on a plane. They usually come in pairs, consisting of an "x" value and a "y" value.
In the Cartesian coordinate system, these are written as \((x, y)\). By looking at coordinates such as \((-1, 2)\) or \((7, 4)\), you can determine the exact position of these points on the graph.
Each coordinate pair represents a point.

• \((-1, 2)\) tells us to move 1 unit left and 2 units up from the origin.
• \((7, 4)\) means move 7 units right and 4 units up.

Having a clear understanding of how to read coordinates is crucial in solving problems related to points and lines on a graph.

What is a Line Segment?

A line segment is a part of a line that connects two distinct points. Unlike a line, a segment has fixed endpoints and does not extend indefinitely in either direction.
The segment between \((-1, 2)\) and \((7, 4)\) represents a direct path between these two points on the graph.

• Endpoints: These are the two points \((-1, 2)\) and \((7, 4)\) in our example.
• Length: A segment's length can be calculated using the distance formula, but here, we're focusing more on the midpoint.

Understanding a line segment helps in grasping how distances and midpoints play vital roles in geometry.

Simplification using the Midpoint Formula

The midpoint formula is a mathematical tool that helps find the center point of a line segment. It works by averaging the coordinates of the endpoints. The formula for finding the midpoint \((M)\) is:\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]For the points \((-1, 2)\) and \((7, 4)\), substituting them into the formula gives:

• \(x\)-coordinate:\(\frac{-1 + 7}{2} = \frac{6}{2}\)
• \(y\)-coordinate:\(\frac{2 + 4}{2} = \frac{6}{2}\)

These calculations simplify to \((3, 3)\), which means \((3, 3)\) is the midpoint. Breaking down these steps and simplifying the equation helps in understanding how averages work in geometry.