Problem 20 - Algebra 1: Concepts and Skills

Question

Find the midpoint of the line segment connecting the given points. \((5,-5),(-5,1)\)

Step-by-Step Solution

Answer

The midpoint of the line segment connecting the points (5, -5) and (-5, 1) is (0, -2).

Step 1: Identify the coordinates of the endpoints.

From the exercise, the coordinates of the given points are (5,-5) and (-5,1).

Step 2: Compute the X-coordinate of the midpoint.

The X-coordinate of the midpoint is given by the average of the X-coordinates of the endpoints. This can be computed using the formula: \[Midpoint_x = (X_1 + X_2)/2\]Substituting the values from the exercise, \[Midpoint_x = (5 + -5)/2 = 0 \]

Step 3: Compute the Y-coordinate of the midpoint.

The Y-coordinate of the midpoint is given by the average of the Y-coordinates of the endpoints. This can be computed using the formula: \[Midpoint_y = (Y_1 + Y_2)/2\]Substituting the values from the exercise, \[Midpoint_y = (-5 + 1)/2 = -2 \]

Step 4: State the coordinates of the midpoint.

Combining the X-coordinate and Y-coordinate computed in steps 2 and 3 respectively, the coordinates of the midpoint are (0, -2).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as coordinate geometry, provides a powerful connection between algebra and geometry. This mathematical field helps us understand points, lines, and planes using a coordinate system. In the Cartesian coordinate system, each point in this system is identified by an ordered pair (x, y) of real numbers.

When considering line segments within this system, such as the one in our exercise, it connects two points in a plane. By knowing each endpoint, you can determine many properties about the line segment, including its length, direction, and midpoint. The midpoint of a line segment is equally distant from both endpoints and is crucial for many geometrical constructions and proofs. Understanding how to compute this is a key aspect of mastering coordinate geometry.

Average Calculation

The midpoint of a line segment involves averaging the coordinates of its endpoints. This relates directly to the concept of the average, a fundamental idea in mathematics used to find a center point among numbers.

In our exercise, we average the x-coordinates and y-coordinates separately to find the midpoint. For any two points \( (X_1, Y_1) \) and \( (X_2, Y_2) \), the formulas used are:

• The average of the x-coordinates gives the x-component of the midpoint:
  \[ Midpoint_x = \frac{X_1 + X_2}{2} \]
• The average of the y-coordinates gives the y-component of the midpoint:
  \[ Midpoint_y = \frac{Y_1 + Y_2}{2} \]

This average calculation finds a balanced center position, making the midpoint equidistant from both endpoints along each axis.

Step by Step Solution

Breaking problems into steps makes them more manageable, especially in mathematics. A step-by-step solution ensures an organized approach and gives clarity to the problem-solving process.

Let's review the steps taken to find the midpoint in our example:

• **Step 1**: Identify the endpoints of the line segment. This is essential for setting up the problem. For our exercise: (5, -5) and (-5, 1).
• **Step 2**: Calculate the average of the x-coordinates: \((5 + -5)/2 = 0\). This finds the x-coordinate of the midpoint.
• **Step 3**: Calculate the average of the y-coordinates: \((-5 + 1)/2 = -2\). This gives the y-coordinate of the midpoint.
• **Step 4**: Combine these results to state the full coordinates of the midpoint: \(0, -2\).

By following these outlined steps, you ensure accuracy in each calculation and gain a deeper understanding of why each component is essential in reaching the solution.