Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 106
Use the distance formula to calculate the distance between the given two points. $$(-5, -2) and (1, -6)$$
Short Answer
Expert verified
The distance is approximately 7.21.
Step by step solution
01
Identify the Points
First, identify the coordinates of the given points. We have two points: \((-5, -2)\) which we will call \( (x_1, y_1) \), and \((1, -6)\) which we will call \( (x_2, y_2) \).
02
Recall the Distance Formula
The formula to calculate the distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
03
Substitute Coordinates into Formula
Substitute \(x_1 = -5\), \(y_1 = -2\), \(x_2 = 1\), and \(y_2 = -6\) into the formula.\[ d = \sqrt{(1 - (-5))^2 + (-6 - (-2))^2} \]
04
Simplify Inside the Parentheses
Perform the calculations inside the parentheses. \(1 - (-5) = 1 + 5 = 6\) and \((-6) - (-2) = -6 + 2 = -4\).
05
Calculate the Squares
Square the results from the previous step. \(6^2 = 36\) and \((-4)^2 = 16\).
06
Add and Take the Square Root
Add the squared values from the previous step and take the square root. \[ d = \sqrt{36 + 16} = \sqrt{52} \] Finally, approximate the square root if needed: \( \sqrt{52} \approx 7.21 \).
07
Conclusion
The distance between the points \((-5, -2)\) and \((1, -6)\) is approximately \(7.21\).
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.
Coordinates
Coordinates are a way to represent points on a plane using two numbers in an ordered pair:
- The first number is called the x-coordinate, which shows how far left or right the point is from the origin.
- The second number is the y-coordinate, indicating the position above or below the origin.
- Point 1 is \((-5, -2)\), where \(-5\) is the x-coordinate and \(-2\) is the y-coordinate.
- Point 2 is \((1, -6)\), with \(1\) as the x-coordinate and \(-6\) as the y-coordinate.
Distance Calculation
Calculating the distance between two points on a plane involves using a specific formula known as the distance formula. This formula helps us measure the straight line distance between two given points \((x_1, y_1)\) and \((x_2, y_2)\).
The distance formula is expressed as:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
This formula takes the difference between the x-coordinates and y-coordinates of the two points:
The distance formula is expressed as:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
This formula takes the difference between the x-coordinates and y-coordinates of the two points:
- Square each difference to ensure they are positive.
- Add these squared differences together.
- Finally, take the square root of this sum to obtain the distance.
Pythagorean Theorem
The distance formula is essentially a practical application of the Pythagorean theorem. This theorem is fundamental in geometry, relating to right-angled triangles:
- According to the Pythagorean theorem: \(a^2 + b^2 = c^2\),
- where \(a\) and \(b\) are the legs of the triangle and \(c\) is the hypotenuse.
- The differences \((x_2 - x_1)\) and \((y_2 - y_1)\) form the legs \(a\) and \(b\).
- The distance \(d\) becomes the hypotenuse \(c\).
Square Root Simplification
Simplifying square roots is a crucial step in finding precise distances. Once we have the squared sum of the coordinate differences, \(\sqrt{52}\) in this exercise, simplifying it helps us get an exact answer:
- The process begins by understanding that \(\sqrt{52}\) can be broken down using prime factorization.
- \(52 = 2 \times 2 \times 13\), so \(\sqrt{52} = \sqrt{4 \times 13} = \sqrt{4} \times \sqrt{13}\).
- Simplifying gives us \(2\sqrt{13}\) as an exact form.
- For practical purposes, we approximate \(\sqrt{52}\) to about \(7.21\), especially if an approximate decimal is more useful.
