Which of the following equations correctly represents the distance between points $(1,-1)$ and $(-2,2)$?
A. $d=\sqrt{(1-2)^2+(2-1)^2}$
B. $d=\sqrt{(1+2)+(-1-2)}$
C. $d=\sqrt{(1-2)^2+(-1+2)^2}$
D. $d=\sqrt{(1+2)^2+(-1-2)^2}$
Answer (1)
Recall the distance formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
Substitute the given points ( 1 , − 1 ) and ( − 2 , 2 ) into the formula: d = ( − 2 − 1 ) 2 + ( 2 − ( − 1 ) ) 2 .
Simplify the expression: d = ( 3 ) 2 + ( − 3 ) 2 = 9 + 9 = 18 .
Identify the correct equation from the options: d = ( 1 + 2 ) 2 + ( − 1 − 2 ) 2 .
Explanation
Problem Analysis and Distance Formula We are given two points, ( 1 , − 1 ) and ( − 2 , 2 ) , and we need to find the equation that correctly represents the distance between them. The distance formula between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by:
d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
Let's analyze the given options and compare them with the distance formula.
Applying the Distance Formula Let's substitute the coordinates of the given points into the distance formula:
x 1 = 1 , y 1 = − 1 x 2 = − 2 , y 2 = 2
d = ( − 2 − 1 ) 2 + ( 2 − ( − 1 ) ) 2 d = ( − 3 ) 2 + ( 3 ) 2 d = ( 1 + 2 ) 2 + ( − 1 − 2 ) 2
Now, let's examine the given options and see which one matches our result.
Comparing with Given Options Now, let's compare the derived equation with the given options:
Option 1: d = ( 1 − 2 ) 2 + ( 2 − 1 ) 2 = ( − 1 ) 2 + ( 1 ) 2 = 1 + 1 = 2 . This is incorrect.
Option 2: d = ( 1 + 2 ) + ( − 1 − 2 ) = 3 + ( − 3 ) = 0 = 0 . This is incorrect.
Option 3: d = ( 1 − 2 ) 2 + ( − 1 + 2 ) 2 = ( − 1 ) 2 + ( 1 ) 2 = 1 + 1 = 2 . This is incorrect.
Option 4: d = ( 1 + 2 ) 2 + ( − 1 − 2 ) 2 = ( 3 ) 2 + ( − 3 ) 2 = 9 + 9 = 18 . This matches our derived equation.
Final Answer The correct equation representing the distance between the points ( 1 , − 1 ) and ( − 2 , 2 ) is:
d = ( 1 + 2 ) 2 + ( − 1 − 2 ) 2
Therefore, the correct answer is the fourth option.
Examples
The distance formula is a fundamental concept in coordinate geometry and has numerous real-world applications. For example, in navigation, you can use the distance formula to calculate the shortest distance between two locations on a map, represented as coordinates on a grid. This is crucial for determining the most efficient route for travel or shipping. Similarly, in computer graphics and game development, the distance formula is used to calculate distances between objects in a virtual environment, enabling realistic interactions and movements.
