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Chapter 10: Problem 94

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ (-4.6,2.1),(-6.7,1.9) $$

Short Answer

Expert verified
The midpoint of the line segment is (-5.65, 2.0).

Step by step solution

01

Understand the Midpoint Formula

The midpoint of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\] This formula finds the average of the x-coordinates and the average of the y-coordinates to determine the midpoint.
02

Identify the Coordinates

Identify the coordinates of the given endpoints: For endpoint \((x_1, y_1)\), we have \((-4.6, 2.1)\).For endpoint \((x_2, y_2)\), we have \((-6.7, 1.9)\).
03

Substitute the Values

Substitute the identified coordinates into the midpoint formula:\[\left( \frac{-4.6 + (-6.7)}{2}, \frac{2.1 + 1.9}{2} \right)\] This will give the coordinates for the midpoint.
04

Calculate the Average of the X-Coordinates

Calculate the average of the x-coordinates: \[\frac{-4.6 + (-6.7)}{2} = \frac{-11.3}{2} = -5.65\]Thus, the x-coordinate of the midpoint is \(-5.65\).
05

Calculate the Average of the Y-Coordinates

Calculate the average of the y-coordinates: \[\frac{2.1 + 1.9}{2} = \frac{4.0}{2} = 2.0\]Thus, the y-coordinate of the midpoint is \(2.0\).
06

Write the Midpoint Coordinates

Combine the x and y midpoint values to write the full midpoint coordinates: The midpoint is \((-5.65, 2.0)\).

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Key Concepts

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

Coordinate Geometry
Coordinate Geometry is a branch of mathematics that deals with geometric figures like points, lines, and shapes on a coordinate plane. It acts like a bridge between algebra and geometry, allowing us to use algebraic methods to solve geometric problems.
By positioning figures on a numerical grid, known as the Cartesian coordinate plane, we can find exact locations of points and calculate distances and midpoints, just like in our original exercise.
  • The plane is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical).
  • Each point is represented by coordinates \( (x, y) \), where \( x \) is the horizontal distance from the origin and \( y \) is the vertical distance.
Thus, in coordinate geometry, every geometric problem can be translated into an algebraic equation, making more complex problems easier to solve.
Line Segment
A line segment is a part of a line that is bounded by two distinct end points. Unlike a line that extends infinitely in both directions, a line segment has a fixed length. In the context of coordinate geometry, line segments are identified by their endpoints, such as the ones given in the exercise.
To analyze a line segment:
  • Determine the endpoints, for example, \((-4.6, 2.1)\) and \((-6.7, 1.9)\) in our exercise.
  • Use these points to calculate properties like midpoints, lengths, and slopes.
Understanding line segments is essential for calculating the midpoint, as they provide the coordinates needed for exact calculations. This enables us to find the exact middle point on the segment by applying the Midpoint Formula.
Averaging
Averaging is a mathematical concept that involves finding the central value or a typical value of a set of numbers. In the calculation of a midpoint, averaging plays a crucial role.
Remember, the Midpoint Formula:\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] involves finding the average of the x-coordinates and y-coordinates of the endpoints.
  • For x-coordinates, add them together and divide by 2 to find the average \( \frac{-4.6 + (-6.7)}{2} \).
  • For y-coordinates, do the same \( \frac{2.1 + 1.9}{2} \).
This simple method of averaging helps find the center point of a line segment by balancing out the coordinates, essentially giving you the midpoint on the plane.

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Most popular questions from this chapter

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