Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 9
Find the distance between \((-2,3)\) and the midpoint of the segment joining \((-2,-2)\) and \((4,3)\).
Short Answer
Expert verified
The distance is \(\frac{\sqrt{61}}{2}\).
Step by step solution
01
Find the Midpoint
To find the midpoint of the segment joining \((-2, -2)\) and \((4, 3)\), use the midpoint formula: \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). Substitute \(x_1 = -2\), \(y_1 = -2\), \(x_2 = 4\), \(y_2 = 3\). This results in: \(\left(\frac{-2 + 4}{2}, \frac{-2 + 3}{2}\right) = \left(1, \frac{1}{2}\right)\). The midpoint is \((1, \frac{1}{2})\).
02
Apply the Distance Formula
Now, find the distance between \((-2, 3)\) and the midpoint \((1, \frac{1}{2})\) using the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Here, \(x_1 = -2\), \(y_1 = 3\), \(x_2 = 1\), \(y_2 = \frac{1}{2}\).
03
Substitute into the Distance Formula
Substitute the values into the distance formula: \(d = \sqrt{(1 - (-2))^2 + \left(\frac{1}{2} - 3\right)^2}\). Simplify: \(d = \sqrt{(1 + 2)^2 + \left(-\frac{5}{2}\right)^2}\). This becomes \(d = \sqrt{3^2 + \left(-\frac{5}{2}\right)^2}\).
04
Calculate the Terms
Calculate the squares: \(3^2 = 9\) and \(\left(-\frac{5}{2}\right)^2 = \frac{25}{4}\). Combine them: \(d = \sqrt{9 + \frac{25}{4}}\).
05
Combine and Simplify
To combine the terms with a common denominator: \(9 = \frac{36}{4}\). So, \(d = \sqrt{\frac{36}{4} + \frac{25}{4}} = \sqrt{\frac{61}{4}}\).
06
Final Simplification
Simplify by taking the square root: \(d = \sqrt{\frac{61}{4}} = \frac{\sqrt{61}}{2}\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Midpoint Formula
To understand the midpoint formula, imagine you have a straight line connecting two points on a coordinate plane. The midpoint is exactly halfway between these two points. It鈥檚 like finding the center point of a segment. The formula is \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]In simpler terms:
- Add the x-coordinates of the two points together and then divide by 2 to find the x-value of the midpoint.
- Add the y-coordinates of the two points together and divide by 2 to find the y-value of the midpoint.
Coordinate Geometry
Coordinate geometry, often called analytic geometry, is the study of geometry using a coordinate system. It is a powerful mathematical tool for describing and analyzing geometric shapes in a plane. Here are some essential ideas:
- Points are represented as coordinates \((x, y)\) on a plane.
- Lines can be described using algebraic equations.
- Shapes and curves can be analyzed using equations and distances between points.
Euclidean Distance
Euclidean distance is the "straight-line" distance between two points in Euclidean space, which is the basic two-dimensional plane used in geometry. When we say "distance," this is typically what we mean. It is calculated using the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]In simple words:
- Subtract the x-coordinates and square the result.
- Repeat for the y-coordinates.
- Add these squares together.
- Take the square root of the sum to find the Euclidean distance.
