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Chapter 7: Problem 68

Find the midpoint of the segment with the given endpoints. $$ (-1,2) \text { and }(1,-3) $$

Short Answer

Expert verified
The midpoint is \( (0, -0.5) \).

Step by step solution

01

Understanding the Midpoint Formula

The midpoint of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) can be found using the formula \[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \].
02

Identify the Coordinates

The coordinates of the endpoints are \( (-1, 2) \) for \((x_1, y_1)\) and \( (1, -3) \) for \((x_2, y_2)\).
03

Apply the Midpoint Formula

Using the identified coordinates in the midpoint formula, calculate each component: \(\frac{-1 + 1}{2} = 0\) for the x-coordinate and \( \frac{2 + (-3)}{2} = -0.5 \) for the y-coordinate.
04

Combine the Components

Thus, the midpoint of the segment is \((0, -0.5)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Midpoint Formula
The Midpoint Formula is a crucial concept in coordinate geometry. It helps find the exact middle point between two designated points on a linear segment. This is incredibly useful for various applications in geometry and algebra.
The formula is given by: \( \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
Using this simple equation, you can compute the average of the x-coordinates and the average of the y-coordinates of the two endpoints.
This provides the coordinates of the midpoint.
For example:
    \((x_1, y_1) = (-1, 2) \) and \((x_2, y_2) = (1, -3) \). Then the midpoint becomes: \( \left( \frac{-1 + 1}{2}, \frac{2 + (-3)}{2} \right) = (0, -0.5)\).

As consistent practice enhances understanding, try utilizing the formula with different sets of coordinates.
Coordinate Geometry
Coordinate Geometry, also known as Analytical Geometry, blends algebra and geometry using a coordinate system.
This system uses coordinates to represent geometric figures like points, lines, and shapes on a plane.

By plotting points using their coordinates and visualizing their relationships, you can solve geometric problems more clearly.
For instance, plotting the points \((-1, 2) \) and \( (1, -3) \) on a graph allows visual inspection of their positions.
This method makes concepts like the midpoint more intuitive.
Coordinate geometry brings powerful tools for
  • Calculating distances between points
  • Finding slopes of lines
  • Identifying shapes
  • Understanding transformations
  • Locating midpoints

By integrating these techniques, it offers extended insight beyond basic geometric principles.
Linear Segments
Linear segments are straight lines connecting two points in a coordinate system.
They鈥檙e fundamental in geometry, representing a direct path from one endpoint to another.
The length, midpoint, and slope of these segments are often topics of interest.

Consider endpoints \((-1, 2) \) and \( (1,-3) \). The segment between them can be analyzed in multiple ways:
    Finding the slope (rate of change between points)
  • Calculating the segment's length (distance formula)
  • Identifying the midpoint (using the Midpoint Formula)
  • Evaluating other geometrical properties

Each point on this line segment satisfies the linear equation derived from these endpoints.
Understanding and calculating different properties of linear segments enhance your grasp on broader coordinate geometry and its applications.

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Most popular questions from this chapter

Find the midpoint of the segment with the given endpoints. $$ \left(\frac{1}{6},-\frac{3}{4}\right) \text { and }\left(-\frac{1}{3}, \frac{5}{6}\right) $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt[3]{x y^{2} z} \sqrt{x^{3} y z^{2}} $$

The absolute value of a complex number \(a+b i\) is its distance from the origin. (See the graph above.) Using the distance formula, we have \(|a+b i|=\sqrt{a^{2}+b^{2}}\) Find the absolute value of each complex number. $$|-3-i|$$

Find the midpoint of the segment with the given endpoints. $$ (4.1,6.9) \text { and }(5.2,-8.9) $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt{x y^{3}} \sqrt[3]{x^{2} y} $$
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