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Chapter 1: Problem 36

Verify the Midpoint Formula by showing the distance between \(P\left(x_{1}, y_{1}\right)\) and \(M\) and the distance between \(M\) and \(Q\left(x_{2}, y_{2}\right)\) are both half of the distance between \(P\) and \(Q\).

Short Answer

Expert verified
The midpoint formula is verified; distances from P to M and M to Q are both half of the distance from P to Q.

Step by step solution

01

Understand the Midpoint Formula

The midpoint of a line segment with endpoints \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is given by the formula: \[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \] This formula calculates the average of the x-coordinates and the y-coordinates of the endpoints.
02

Calculate the Distance from P to Q

Use the distance formula to find the distance between the points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\). The distance formula is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
03

Calculate the Distance from P to M

Substitute the midpoint \(M\) into the distance formula to find the distance from \(P(x_1, y_1)\) to \(M\): \[ d_{PM} = \sqrt{\left(\frac{x_1 + x_2}{2} - x_1\right)^2 + \left(\frac{y_1 + y_2}{2} - y_1\right)^2} \] Simplify the expression to find \[ d_{PM} = \frac{1}{2} \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This shows that the distance from \(P\) to \(M\) is half of the distance from \(P\) to \(Q\).
04

Calculate the Distance from M to Q

Similarly, use the midpoint \(M\) in the distance formula to find the distance from \(M\) to \(Q(x_2, y_2)\): \[ d_{MQ} = \sqrt{\left(\frac{x_1 + x_2}{2} - x_2\right)^2 + \left(\frac{y_1 + y_2}{2} - y_2\right)^2} \] Simplify to get \[ d_{MQ} = \frac{1}{2} \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This shows that the distance from \(M\) to \(Q\) is also half of the distance from \(P\) to \(Q\).
05

Conclusion on the Midpoint Formula Verification

Since both \(d_{PM}\) and \(d_{MQ}\) are equal to \(\frac{1}{2}\) of the distance between \(P\) and \(Q\), we have verified that the midpoint formula divides the segment into two equal halves.

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Key Concepts

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

Distance Formula
The distance formula is a vital tool in coordinate geometry used to find the length of a line segment connecting two points in a plane. If you have two points, \( P(x_1, y_1) \) and \( Q(x_2, y_2) \), the distance \( d \) is calculated using their coordinates:
  • First, find the difference between the \( x \)-coordinates: \( x_2 - x_1 \).
  • Then, find the difference between the \( y \)-coordinates: \( y_2 - y_1 \).
  • Apply the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Using this formula, you can measure directly between any two points, providing a concrete way to handle spatial problems. This formula comes from the Pythagorean theorem, reflecting the idea that any distance in a rectangular coordinate system can be viewed as the hypotenuse of a right triangle.
Line Segment
A line segment in geometry is part of a line that has two distinct endpoints. Unlike a full line, it does not extend infinitely in either direction. When you have a line segment defined by two points, such as \( P \) and \( Q \), it is often denoted as segment \( PQ \). Key characteristics include:
  • It has a fixed length, which can be computed using the distance formula.
  • It is the simplest form of geometry defined by points, connecting them directly without deviation.
  • In exercises like the Midpoint Formula verification, a line segment is crucial for determining how points are situated relative to each other.
Understanding line segments helps in solving problems related to bisecting or dividing lengths. This concept is foundational to Euclidean geometry, where you often need to establish relationships between different geometric figures.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, bridges algebra and geometry. It utilizes a coordinate system to define geometric shapes like points, lines, and curves. The importance of coordinate geometry for solving geometric problems includes:
  • Facilitating the computation of distances and midpoints using algebraic methods.
  • Allowing for precise descriptions of geometric figures based on their positions.
  • Enabling deeper exploration into relationships and proofs of geometric properties using equations.
With coordinate geometry, you can transition from abstract, theoretical descriptions to practical applications and visualizations on axes. It allows you to see equations and geometric configurations come to life, providing a solid grounding for solving problems like finding the midpoint of line segments.
Mathematical Proofs
Mathematical proofs form the backbone of verifying formulas and theorems in geometry. When you prove a formula like the midpoint formula, you confirm its universality and correctness. Here's how mathematical proofs help:
  • They ensure that mathematical statements are logically sound and reliable across all cases.
  • Proofs provide a step-by-step reasoning to establish why a formula is true, such as demonstrating equal distances in the Midpoint Formula exercise.
  • They deepen understanding by showing where a formula originates from and how different mathematical concepts interconnect.
Using proofs, mathematicians provide assurance of the solidity of mathematical theories. This helps in advancing further study and application, giving students confidence in their use of formulas and tools.

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Most popular questions from this chapter

In Exercises \(21-41,\) determine analytically if the following functions are even, odd or neither. $$ f(x)=2 x^{3}-x $$

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