Problem 88 - Algebra A Combined Function

Question

Find the midpoint of each line segment whose endpoints are given. (3,9)\(;(7,11)\)

Step-by-Step Solution

Answer

The midpoint is (5, 10).

Step 1: Identify the Coordinates

The endpoints of the line segment are given as two points: \((3,9)\) and \((7,11)\). Let the first point \((3,9)\) have coordinates \((x_1, y_1)\) and the second point \((7,11)\) have coordinates \((x_2, y_2)\).

Step 2: Use the Midpoint Formula

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

Step 3: Calculate the Midpoint's x-coordinate

Substitute \(x_1 = 3\) and \(x_2 = 7\) into the formula for the x-coordinate of the midpoint:\[ x = \frac{3 + 7}{2} = \frac{10}{2} = 5 \]

Step 4: Calculate the Midpoint's y-coordinate

Substitute \(y_1 = 9\) and \(y_2 = 11\) into the formula for the y-coordinate of the midpoint:\[ y = \frac{9 + 11}{2} = \frac{20}{2} = 10 \]

Step 5: Combine the Coordinates

Combine the calculated x-coordinate and y-coordinate to form the midpoint:\( (5, 10) \).

Key Concepts

Understanding Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that allows you to describe the location of points in space using ordered pairs (or triplets in three dimensions). Points are placed on a coordinate plane formed by a horizontal x-axis and a vertical y-axis. Each point has a unique position described by two numerical coordinates, usually written in the format (x, y).
Clearning up the confusion about coordinates can be crucial. The first number in each pair, called the x-coordinate, specifies the point's horizontal position. The second number, the y-coordinate, states the vertical position. Together, these numbers unlock the power of precise mathematical descriptions of geometric figures.

• Example: The point (3, 9) is located 3 units to the right along the x-axis, and 9 units up along the y-axis.

Understanding how to use these coordinates to define and calculate geometric properties is vital in mathematics and many real-world applications such as engineering and computer graphics.

Exploring Line Segments

A line segment is a fundamental concept in geometry. It's essentially a part of a line bounded by two distinct endpoints. Unlike a line, a line segment does not extend infinitely, making it a defined measurable path.
To visualize a line segment, imagine a straight stick. It has a beginning and an end, which in our case are defined by coordinates on the coordinate plane.
For example, the line segment we're working with has endpoints at (3, 9) and (7, 11). These points determine both the length and location of the segment within the coordinate system.

• Connecting Points: These create the segment by linking the points directly.
• Real-world connection: Think of it as the shortest path between two locations.

When calculating or describing line segments, acknowledging their endpoints is crucial, just like naming streets with start and end locations.

Applying Calculation Techniques

When it comes to calculation techniques in coordinate geometry, applying formulas with precision can reveal the necessary solutions. For finding midpoints, the midpoint formula is an essential tool. It's like finding the center of balance between two endpoints.
The midpoint formula is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]where \((x_1, y_1)\) are the coordinates of the first endpoint, and \((x_2, y_2)\) are those of the second endpoint. This formula gives you a new point that is equidistant from the two endpoints along both axes.

• Step-by-step Use: Substitute the known coordinates into the formula to find the midpoint.
• Check your work: Ensure your calculated midpoint is logically placed between the original points.

For our exercise, by substituting \(x_1 = 3\), \(x_2 = 7\), \(y_1 = 9\), and \(y_2 = 11\) into the formula, we found the midpoint to be \( (5, 10) \). Reinforce these skills through practice to gain confidence in geometric calculations.