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Chapter 1: Problem 42

Find the coordinates of the midpoint of a segment having the given endpoints. $$L(-1.4,3.2), M(2.6,-5.4)$$

Short Answer

Expert verified
The midpoint coordinates are (0.6, -1.1).

Step by step solution

01

Identify the coordinates of endpoints

We begin by identifying the given endpoints of the segment: - Endpoint \( L = (-1.4, 3.2) \) - Endpoint \( M = (2.6, -5.4) \).
02

Apply the midpoint formula

The midpoint \( (x_m, y_m) \) of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[(x_m, y_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
03

Substitute the endpoint values into the formula

Substitute the coordinates of endpoints \(L\) and \(M\) into the midpoint formula: \[(x_m, y_m) = \left( \frac{-1.4 + 2.6}{2}, \frac{3.2 + (-5.4)}{2} \right)\]
04

Perform the calculations

Calculate the values for the midpoint coordinates: \[x_m = \frac{-1.4 + 2.6}{2} = \frac{1.2}{2} = 0.6 \]\[y_m = \frac{3.2 - 5.4}{2} = \frac{-2.2}{2} = -1.1 \]
05

Write the final coordinates of the midpoint

The coordinates of the midpoint of the segment are \((0.6, -1.1)\).

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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 uses numbers to represent points, lines, and shapes on a plane. This powerful field allows you to analyze geometric shapes numerically and understand the relationships between different geometric entities.
  • Points are expressed using pairs of numbers, typically written as \( (x, y) \). These numbers, known as coordinates, represent a specific location on the plane.
  • The plane itself is divided by a horizontal axis (x-axis) and a vertical axis (y-axis), which helps in locating the points accurately.
  • Lines are described algebraically using equations, and the slope is often used to determine their steepness and direction.
Understanding coordinate geometry enables you to solve complex geometry problems by using algebraic techniques, making it easier to find distances between points and calculate midpoints.
Endpoints
Endpoints refer to the two points that mark the boundary of a line segment. These points are crucial since they define the exact length and position of the line on the coordinate plane.
  • The given problem provides us with two endpoints: \(L = (-1.4, 3.2) \) and \(M = (2.6, -5.4) \).
  • Endpoints are denoted as \( (x_1, y_1) \) and \( (x_2, y_2) \), which are used in various formulas like the midpoint formula.
  • Line segments are finite and differ from lines since they can't extend infinitely; they are specifically defined by their endpoints.
Once you have identified the endpoints, you can easily apply measurement formulas to find distances and midpoints.
Calculating Midpoints
Calculating the midpoint of a line segment involves finding the point that is equidistant from both endpoints. This midpoint divides the segment into two equal parts.
The formula to find the midpoint \( (x_m, y_m) \) of a segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is:\[(x_m, y_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]In our exercise, the endpoints are \(L = (-1.4, 3.2) \) and \(M = (2.6, -5.4) \). Here's how you calculate it:
  • Substitute the x-coordinates: \( x_m = \frac{-1.4 + 2.6}{2} = 0.6 \)
  • Substitute the y-coordinates: \( y_m = \frac{3.2 - 5.4}{2} = -1.1 \)
Therefore, the midpoint of this segment is \( (0.6, -1.1) \). This concept is essential not only in geometry but also in various practical applications like computer graphics and navigation.

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