Problem 21 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The midpoint of the line segment connecting the points \((-1,1)\) and \((-4,-4)\) is \((-2.5,-1.5)\)

Step 1: Identify the coordinates

The coordinates of the first point is \((-1,1)\), hence \(x_1 = -1\) and \(y_1 = 1\). The coordinates of the second point is \((-4,-4)\), hence \(x_2 = -4\) and \(y_2 = -4\)

Step 2: Substitute into the midpoint formula

Substitute \(x_1\), \(x_2\), \(y_1\) and \(y_2\) into the midpoint formula that is \((\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\). You will obtain \((\frac{-1+-4}{2}, \frac{1+-4}{2})\)

Step 3: Simplify the expressions

Simplify the expressions in the brackets. For the x-coordinate, add -1 and -4 to get -5, then divide -5 by 2 to get -2.5. For the y-coordinate, add 1 and -4 to get -3, then divide -3 by 2 to get -1.5. Thus, the result is \((-2.5,-1.5)\)

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that allows us to describe the position and shape of geometric figures using a coordinate system. This system typically uses an ordered pair of numbers known as coordinates. Each coordinate corresponds to a position on the horizontal (x-axis) and the vertical (y-axis) axes on a Cartesian plane.

In this context, every point in coordinate geometry is represented with a pair of numbers: \(x, y\). The first number is the x-coordinate, indicating the horizontal position, and the second number is the y-coordinate, indicating the vertical position.

Understanding how to read and interpret these coordinates is essential for solving problems involving geometric shapes and lines. They help us trace and analyze line segments, angles, and even areas of complex figures, providing a crucial foundation for applications in physics, engineering, and computer science.

Line Segments

A line segment is one of the simplest concepts in geometry. It consists of two endpoints and includes all the points between them. Unlike a line that extends infinitely in both directions, a line segment is finite and has a definite beginning and end.

When we're tasked with finding the midpoint of a line segment, it means we need to locate the exact center point that lies equidistant from both endpoints. This involves calculating the average of the x-coordinates and the y-coordinates of the endpoints determined by their positions in a Cartesian plane.

Understanding line segments is crucial for a variety of mathematical problems and real-life applications, such as architectural design, construction, and computer graphics, where precise measurements are needed.

Midpoint Calculation

The midpoint of a line segment is the point that divides the segment into two equal parts. To find this midpoint, we use a simple yet powerful tool known as the midpoint formula:

\[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]

This formula computes the average of the x-coordinates and the y-coordinates of the endpoints of the line segment. For example, given points \((-1, 1)\) and \((-4, -4)\), we substitute into the formula:

• \(x\)-coordinate: \( \frac{-1 + (-4)}{2} = -2.5 \)
• \(y\)-coordinate: \( \frac{1 + (-4)}{2} = -1.5 \)

Thus, the midpoint is \((-2.5, -1.5)\).

This calculation is fundamental not just in mathematics, but also in real-world applications such as computer graphics, navigation systems, and geographical mapping, where determining intermediate points accurately is essential.