Which three ordered pairs describe points that are 5 units away from [tex]P (-1,2)[/tex]?
[tex](2,-2)[/tex]
[tex](-6,7)[/tex]
[tex](-1,-3)[/tex]
[tex](1,5)[/tex]
[tex](4,2)[/tex]
Answer (2)
Calculate the distance between point P ( − 1 , 2 ) and each of the given points using the distance formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
Find that the distance between P and ( 2 , − 2 ) is 5.
Find that the distance between P and ( − 1 , − 3 ) is 5.
Find that the distance between P and ( 4 , 2 ) is 5.
The three ordered pairs that are 5 units away from P ( − 1 , 2 ) are: ( 2 , − 2 ) , ( − 1 , − 3 ) , ( 4 , 2 ) .
Explanation
Problem Analysis and Distance Formula We are given the point P ( − 1 , 2 ) and asked to find which of the given points are 5 units away from P . We will use the distance formula to calculate the distance between P and each of the given points. 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
Calculate Distances
Point (2, -2): d = ( 2 − ( − 1 ) ) 2 + ( − 2 − 2 ) 2 = ( 3 ) 2 + ( − 4 ) 2 = 9 + 16 = 25 = 5
Point (-6, 7): d = ( − 6 − ( − 1 ) ) 2 + ( 7 − 2 ) 2 = ( − 5 ) 2 + ( 5 ) 2 = 25 + 25 = 50 = 5 2 ≈ 7.07
Point (-1, -3): d = ( − 1 − ( − 1 ) ) 2 + ( − 3 − 2 ) 2 = ( 0 ) 2 + ( − 5 ) 2 = 0 + 25 = 25 = 5
Point (1, 5): d = ( 1 − ( − 1 ) ) 2 + ( 5 − 2 ) 2 = ( 2 ) 2 + ( 3 ) 2 = 4 + 9 = 13 ≈ 3.61
Point (4, 2): d = ( 4 − ( − 1 ) ) 2 + ( 2 − 2 ) 2 = ( 5 ) 2 + ( 0 ) 2 = 25 + 0 = 25 = 5
Identify Points at Distance 5 From the calculations above, we see that the points (2, -2), (-1, -3), and (4, 2) are 5 units away from the point P ( − 1 , 2 ) .
Final Answer The three ordered pairs that are 5 units away from P ( − 1 , 2 ) are ( 2 , − 2 ) , ( − 1 , − 3 ) , and ( 4 , 2 ) .
Examples
Understanding distance calculations is crucial in various real-world applications. For instance, when planning a city layout, architects and urban planners use the distance formula to determine the optimal placement of buildings, parks, and other amenities. They consider factors like walking distances, transportation routes, and the overall accessibility of different locations to create a well-connected and efficient urban environment. Similarly, in logistics and delivery services, calculating distances between multiple destinations helps optimize routes, minimize travel time, and reduce fuel consumption, leading to cost savings and improved service efficiency.
The points that are 5 units away from P ( − 1 , 2 ) are (2, -2), (-1, -3), and (4, 2).
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