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Chapter 3: Problem 57

Find the midpoint of each segment with the given endpoints. $$ (-8,4) \text { and }(-2,-6) $$

Short Answer

Expert verified
The midpoint is \((-5, -1)\).

Step by step solution

01

Identifying the Formula for Midpoint

The formula for finding the midpoint \((x_m, y_m)\) of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ x_m = \frac{x_1 + x_2}{2} \]\[ y_m = \frac{y_1 + y_2}{2} \]
02

Substituting the Coordinates

Substitute the given endpoints \((-8, 4)\) and \((-2, -6)\) into the formula:\[ x_m = \frac{-8 + -2}{2} \]\[ y_m = \frac{4 + -6}{2} \]
03

Simplifying the Expressions

Perform the arithmetic operations to simplify the expressions:\[ x_m = \frac{-10}{2} = -5 \]\[ y_m = \frac{-2}{2} = -1 \]
04

Writing the Final Midpoint Coordinates

Combine the simplified values to write the coordinates of the midpoint:The midpoint is \((-5, -1)\).

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Key Concepts

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

Understanding Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a mathematical framework that uses a coordinate system to help solve geometric problems. In this system, every point is defined by a pair of numerical coordinates. These coordinates correspond to the point's position along two perpendicular axes鈥攖he X-axis (horizontal) and the Y-axis (vertical).

This method is crucial for solving problems involving distances, midpoints, slopes, and shapes. By using coordinates, complex geometric concepts can be understood and calculated systematically.
What Are Endpoints?
Endpoints are the points located at the ends of a line segment. For example, in the given problem, \((-8, 4)\) and \((-2, -6)\) are endpoints.

Identifying the coordinates of endpoints allows us to use various formulas in coordinate geometry, like the midpoint formula. This formula finds the exact center point between two endpoints.
  • The points \( A \) and \( B \) are used to calculate the midpoint.
  • Endpoints serve as references in setting up and solving geometric problems.
Performing Arithmetic Operations
Arithmetic operations include addition, subtraction, multiplication, and division. These basic mathematical procedures are applied in the given problem to find the midpoint.

When performing these operations with coordinates, it鈥檚 important to keep track of each step:
  • First, add the X-coordinates of the endpoints together.
  • Then, add the Y-coordinates together.
  • Finally, divide each sum by 2 to find the average.
These steps ensure that we correctly find the midpoint using the provided formula.
Simplifying Expressions
Simplifying expressions is a fundamental step in solving math problems. It involves reducing an equation or expression to its simplest form.

In the given problem, we simplified the expressions for the X and Y coordinates:
  • Calculated \( x_m \) using \( x_m = \frac{-8 + -2}{2} \) which simplified to \( x_m = -5 \).
  • Calculated \( y_m \) using \( y_m = \frac{4 + -6}{2} \) which simplified to \( y_m = -1 \).
Simplification makes it easier to combine the results and find the final coordinates of the midpoint.

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Most popular questions from this chapter

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ x-3 y=0 $$

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ -\frac{2}{3} y=x $$

If a line has slope \(-\frac{4}{9},\) then any line parallel to it has slope _____, and any line perpendicular to it has slope _____.

An equation that defines \(y\) as a function fof \(x\) is given. (a) Solve for \(y\) in terms of \(x,\) and \(r e-\) place \(y\) with the function notation \(f(x) .\) (b) Find \(f(3) .\) See Example 6. $$ y-3 x^{2}=2 $$

Use your knowledge of the slopes of parallel and perpendicular lines. Is the figure with vertices at \((-11,-5),(-2,-19),(12,-10),\) and \((3,4)\) a parallelogram? Is it a rectangle? (Hint: A rectangle is a parallelogram with a right angle.)
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