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Chapter 10: Problem 22

Find the coordinates of the midpoint of each segment. \(\overline{A B}\), with endpoints \(A(-2,6)\) and \(B(8,3)\)

Short Answer

Expert verified
Midpoint of segment \( \overline{AB} \) is \( (3, 4.5) \).

Step by step solution

01

Identify Endpoint Coordinates

The problem asks for the midpoint of segment \( \overline{AB} \). The given endpoints are \( A(-2,6) \) and \( B(8,3) \). Note these coordinates: \( x_1 = -2 \), \( y_1 = 6 \), \( x_2 = 8 \), and \( y_2 = 3 \).
02

Use the Midpoint Formula

The midpoint formula is given by \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). Use the coordinates of points \( A \) and \( B \) in this formula to find the midpoint.
03

Calculate Midpoint for x-coordinate

Substitute \( x_1 \) and \( x_2 \) into the midpoint formula: \( \frac{-2 + 8}{2} \). Compute to find the x-coordinate of the midpoint. \( \frac{6}{2} = 3 \).
04

Calculate Midpoint for y-coordinate

Substitute \( y_1 \) and \( y_2 \) into the midpoint formula: \( \frac{6 + 3}{2} \). Compute to find the y-coordinate of the midpoint. \( \frac{9}{2} = 4.5 \).
05

Combine Calculations into Midpoint Coordinates

The calculations give you the coordinates of the midpoint of \( \overline{AB} \), which are \( (3, 4.5) \). Therefore, the midpoint is \( (3, 4.5) \).

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Key Concepts

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

Coordinates
Coordinates are numerical values used to define the position of a point on a plane. In the realm of geometry, particularly in the Cartesian coordinate system, a coordinate pair is used to represent any point. This system divides the plane with a horizontal axis (x-axis) and a vertical axis (y-axis).
For example, let's consider point A(-2, 6) on a plane:
  • -2 is the x-coordinate, specifying its horizontal position.
  • 6 is the y-coordinate, indicating its vertical position.
Similarly, for point B(8, 3), the coordinates are:
  • 8 as the x-coordinate.
  • 3 as the y-coordinate.
This understanding of coordinates is fundamental when using formulas, like the Midpoint Formula, to find the middle of segments or the distance between points on a plane.
Segment
A segment specifically refers to the part of a line that is bounded by two distinct endpoints. Think of it like a piece of string cut from a longer line. Unlike a line that stretches infinitely, a segment only covers the finite distance between its endpoints.
In our case, segment \( \overline{AB} \) is the finite line segment joining points A(-2, 6) and B(8, 3). Here's what you need to know about segments:
  • They have a definite length, which can be calculated using the distance formula.
  • They serve as a basis for finding a midpoint, which is a point exactly halfway along the segment.
Understanding segments is crucial because they lay the groundwork for delving deeper into concepts like midpoints and line equations.
Endpoint Calculation
The endpoint calculation is a key part of finding the midpoint of a segment. By calculating the midpoint, you find the point that's equidistant from both endpoints鈥攅ssentially the 'average' position between them on the plane.
The Midpoint Formula we use is:\[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]For \( \overline{AB} \), the calculation steps are as follows:
  • Substitute the x-coordinates: \( x_1 = -2 \) and \( x_2 = 8 \) into the formula to get the x-coordinate of the midpoint: \( \frac{-2 + 8}{2} = 3 \).
  • Substitute the y-coordinates: \( y_1 = 6 \) and \( y_2 = 3 \) into the formula to find the y-coordinate: \( \frac{6 + 3}{2} = 4.5 \).
Thus, the midpoint coordinates are (3, 4.5). These steps show that calculating a segment's midpoint involves a straightforward process of averaging the respective coordinates of its endpoints.

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