Find the distance when [tex]$y_2=9, y_1=13, x_2=4$[/tex], and [tex]$x_1=0$[/tex]. Use the following formula: distance between points on a coordinate plane [tex]$=\sqrt{ }\left(y_2-y_1\right)^2+\left(x_2-x_1\right)^2$[/tex]
A. 5.66
B. 5.36
C. 4.12
D. 6.95
Answer (2)
Substitute the given coordinates y 2 = 9 , y 1 = 13 , x 2 = 4 , and x 1 = 0 into the distance formula: ( 9 − 13 ) 2 + ( 4 − 0 ) 2 .
Simplify the expression inside the square root: ( − 4 ) 2 + ( 4 ) 2 = 16 + 16 = 32 .
Simplify the square root: 32 = 4 2 .
Approximate the value: 4 2 ≈ 5.66 . Therefore, the distance between the points is 5.66 .
Explanation
Problem Analysis and Given Data We are given the distance formula between two points on a coordinate plane as:
( y 2 − y 1 ) 2 + ( x 2 − x 1 ) 2
We are also given the coordinates of the two points: y 2 = 9 , y 1 = 13 , x 2 = 4 , and x 1 = 0 . Our goal is to find the distance between these two points.
Substitute Values into the Formula To find the distance, we will substitute the given values into the distance formula:
( 9 − 13 ) 2 + ( 4 − 0 ) 2
Now, we simplify the expression inside the square root.
Simplify the Expression First, we calculate the differences within the parentheses:
( − 4 ) 2 + ( 4 ) 2
Next, we square each of these differences:
16 + 16
Then, we add the squared values:
32
Finally, we simplify the square root.
Calculate the Final Distance We can simplify 32 by factoring out the largest perfect square. Since 32 = 16 × 2 , we have:
32 = 16 × 2 = 16 × 2 = 4 2
Now, we approximate the value of 4 2 . Since 2 ≈ 1.414 , we have:
4 2 ≈ 4 × 1.414 = 5.656
Comparing this to the given options, the closest answer is 5.66.
Final Answer Therefore, the distance between the points is approximately 5.66.
Examples
The distance formula is a fundamental concept in coordinate geometry and has numerous real-world applications. For instance, imagine you are using a GPS navigation system to find the distance between two locations. The GPS uses coordinates (latitude and longitude) to pinpoint locations on a map, and the distance formula helps calculate the straight-line distance between these points. This calculation is crucial for estimating travel time and planning routes. Similarly, in fields like urban planning or logistics, the distance formula can be used to optimize the placement of facilities or the routes of delivery vehicles, minimizing travel distances and improving efficiency.
Using the distance formula, the distance between the two points is calculated to be approximately 5.66, which matches option A. This was found by substituting the given coordinates into the formula and simplifying. Therefore, the correct choice is 5.66.
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