Problem 92 - Algebra A Combined Function

Question

Find the midpoint of each line segment whose endpoints are given. (-2,5)\(;(2,6)\)

Step-by-Step Solution

Answer

The midpoint of the line segment is (0, 5.5).

Step 1: Identify the Endpoints

The endpoints of the line segment are given as \((-2, 5)\) and \((2, 6)\). Our task is to find the midpoint of this line segment.

Step 2: Use the Midpoint Formula

The formula to find the midpoint \(M(x, y)\) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is \( M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).

Step 3: Calculate the Midpoint's x-Coordinate

Substitute \(x_1 = -2\) and \(x_2 = 2\) into the formula. The x-coordinate of the midpoint is \(\frac{-2 + 2}{2} = \frac{0}{2} = 0\).

Step 4: Calculate the Midpoint's y-Coordinate

Substitute \(y_1 = 5\) and \(y_2 = 6\) into the formula. The y-coordinate of the midpoint is \(\frac{5 + 6}{2} = \frac{11}{2} = 5.5\).

Step 5: Write the Midpoint

Combine the coordinates to identify the midpoint: \((0, 5.5)\). This is the midpoint of the line segment.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where we use algebra to study geometrical objects and their properties through a coordinate system. It allows us to represent geometric figures such as points, lines, and shapes using numerical coordinates. For example, each point is defined by its position on a plane via an ordered pair of numbers, \( (x, y) \). This system makes it convenient to perform calculations relating to distance, slope, and intersections between various geometric figures.

**Key Benefits of Coordinate Geometry**

• Bridges algebra with geometry, allowing us to solve geometric problems using algebraic equations.
• Makes visualization of complex geometric shapes easier by providing a straightforward method to graph these figures on a plane.
• Helps deduce properties of geometric figures and solve real-world problems involving space and form.

Coordinate geometry is fundamentally about understanding how the algebraic representation of points and lines contributes to our grasp of spatial relationships.

Midpoint Formula

The midpoint formula is a simple yet powerful tool in coordinate geometry used to find the center point or midpoint of a line segment. Given two endpoints, the formula determines the point that lies exactly halfway between them.

The formula for the midpoint \( M(x, y) \) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is:\[M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]
This formula essentially averages the x-coordinates and y-coordinates of the endpoints.

**Steps to Use the Midpoint Formula**

• Take the x-coordinates of the two endpoints, add them together, and divide by 2. This gives the x-coordinate of the midpoint.
• Do the same with the y-coordinates of the endpoints to get the y-coordinate of the midpoint.
• The combination of these two results is the midpoint coordinate \((M_x, M_y)\).

Understanding the midpoint formula is crucial as it can be used in various applications where determining the center of a line segment is needed.

Distance Between Points

In the realm of coordinate geometry, the distance between two points refers to the length of the straight line segment that links them. Calculating this distance is essential for various applications, such as analyzing and comparing geometric shapes and figures.

The distance formula, which comes directly from the Pythagorean theorem, calculates the space between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a plane. The formula is:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
This formula computes the "straight-line" or Euclidean distance, which is the most common way to measure distance in coordinate geometry.

**Steps to Calculate Distance Using the Formula**

• Subtract the x-coordinate of the second point from the x-coordinate of the first point, then square the result.
• Do the same for the y-coordinates.
• Add these squared values and take the square root of the sum to get the distance \(d\).

By understanding how to compute distance, students can better analyze the spatial relationships between points and other geometric entities.

Line Segments

A line segment is a part of a line that is bounded by two distinct endpoints. In contrast to a line, which extends infinitely, a line segment has a specific start and end point, making it a finite piece of a line. This concept is fundamental in geometry and is vital for understanding shapes, sizes, and the properties of figures.

**Properties of Line Segments**

• Defined by two endpoints that are fixed and do not change unless altered by outside forces.
• The length of a line segment can be measured using the distance formula to determine the precise distance between its endpoints.
• Can serve as sides of polygons, diagonals in shapes, and other structures in geometric figures.

Being one of the simplest geometrical units, line segments are involved in various ways as building blocks of more complex shapes and figures. Their study helps in understanding the fundamentals of geometry and laying the groundwork for more complex geometric analyses.