Find the distance between the pair of points [tex]$\left(\frac{9}{2}, \frac{1}{5}\right)$[/tex] and [tex]$\left(\frac{1}{2}, \frac{6}{5}\right)$[/tex]. If necessary, express the answer in simplified radical form and then round to two decimal places.
The distance between the given points is $\square$ units.
(Simplify your answer. Type an exact answer, using radicals as needed)
Answer (1)
Use the distance formula d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
Substitute the coordinates of the points ( 2 9 , 5 1 ) and ( 2 1 , 5 6 ) into the formula.
Simplify the expression to find the distance in simplified radical form: d = 17 .
Approximate the value to two decimal places: 4.12 .
Explanation
Problem Analysis We are given two points ( 2 9 , 5 1 ) and ( 2 1 , 5 6 ) . Our goal is to find the distance between these two points, express the answer in simplified radical form, and then round to two decimal places.
Distance Formula We will use the distance formula to calculate the distance d between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) , which is given by:
d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
In our case, ( x 1 , y 1 ) = ( 2 9 , 5 1 ) and ( x 2 , y 2 ) = ( 2 1 , 5 6 ) .
Substitution and Simplification Substitute the given coordinates into the distance formula:
d = ( 2 1 − 2 9 ) 2 + ( 5 6 − 5 1 ) 2
Simplify the expression inside the parentheses:
d = ( − 2 8 ) 2 + ( 5 5 ) 2
d = ( − 4 ) 2 + ( 1 ) 2
Calculating Squares Calculate the squares:
d = 16 + 1
d = 17
Simplified Radical Form and Approximation The distance in simplified radical form is 17 units. Now, we need to approximate the value of 17 to two decimal places. The result of this operation is approximately 4.12.
Final Answer Therefore, the distance between the points ( 2 9 , 5 1 ) and ( 2 1 , 5 6 ) is 17 units, which is approximately 4.12 units when rounded to two decimal places.
Examples
The distance formula is a fundamental concept in coordinate geometry and has numerous real-world applications. For instance, imagine you're using a GPS to navigate in a city. The GPS uses coordinates to locate your current position and the location of your destination. The distance formula is then used to calculate the shortest distance between these two points, providing you with an estimate of how far you need to travel. This principle is also used in various fields like surveying, aviation, and even video game development to calculate distances between objects or characters in a virtual environment.
