Problem 5 - Algebra 1: Concepts and Skills

Question

Find the midpoint of the line segment with the given endpoints. \((-5,3),(-3,-3)\)

Step-by-Step Solution

Answer

The midpoint of the segment with endpoints \((-5,3)\) and \((-3,-3)\) is \((-4, 0)\)

Step 1: Assign Coordinates

Assign \((-5, 3)\) as \((x_1, y_1)\) and \((-3, -3)\) as \((x_2, y_2)\).

Step 2: Apply the Midpoint Formula for X-Coordinate

Apply the formula \(\frac{x_1 + x_2}{2}\) to find the x-coordinate of the midpoint. Substituting the values, we get \(\frac{-5 + -3}{2} = -4\)

Step 3: Apply the Midpoint Formula for Y-Coordinate

Apply the formula \(\frac{y_1 + y_2}{2}\) to find the y-coordinate of the midpoint. Substituting the values, we get \(\frac{3 + -3}{2} = 0\)

Step 4: Result

Combine the x and y coordinates of the midpoint calculated in the previous steps to find the entire midpoint.

Key Concepts

Coordinates

Coordinates are numbers that represent a point's position on a grid or plane. In a two-dimensional plane, coordinates are given in pairs, which consist of an x-coordinate and a y-coordinate. These coordinates help us identify the exact location of a point on a plane.
For example, if you have a point labeled (-5, 3), ti denotes that the point is 5 units to the left of the origin on the x-axis and 3 units up on the y-axis. Similarly, the point (-3, -3) tells us that it is positioned 3 units to the left of the origin and 3 units down on the y-axis.
Understanding coordinates is essential for solving any problems involving geometric concepts like line segments, shapes, and graphs.

• The x-coordinate is the horizontal position.
• The y-coordinate is the vertical position.
• Coordinates are always written in brackets, e.g., (x, y).

Line Segment

A line segment is a part of a line that is bounded by two distinct end points. Unlike a line, which extends infinitely in both directions, a line segment has a fixed length.
In the original exercise, a line segment connects the points (-5, 3) and (-3, -3). This is the section of a line that exists between these two points.
Line segments are essential in geometry because they represent the shortest path between two points. The concept is fundamental in constructing shapes and analyzing distances. It's a basic building block of many geometric figures.

• A line segment has definite starting and ending points, known as endpoints.
• You can calculate the length of a line segment using distance formulas.
• Midpoints divide the segment into two equal halves.

Endpoints

Endpoints are the points that mark the ends of a line segment or a ray. They are the final or starting positions of the line segment, and identifying them is key to analyzing or manipulating the segment.
In our example, the endpoints of the line segment are (-5, 3) and (-3, -3). Starting by identifying these endpoints is often the first step in solving problems related to line segments, like finding a midpoint.
Endpoints play a crucial role in geometry as they help you define the scope and measure the properties of the line segment.

• Endpoints are necessary for calculating midpoints.
• They are used to determine the length of a segment.
• In plotting graphs, endpoints help in determining direction or limits.