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](1,5)[/tex]
[tex](-1,-3)[/tex]
[tex](4,2)[/tex]
Answer (1)
Use the distance formula to calculate the distance between point P ( − 1 , 2 ) and each of the given points.
Calculate the distance between P ( − 1 , 2 ) and ( 2 , − 2 ) : d = 5 .
Calculate the distance between P ( − 1 , 2 ) and ( − 6 , 7 ) : d = 5 2 .
Calculate the distance between P ( − 1 , 2 ) and ( 1 , 5 ) : d = 13 .
Calculate the distance between P ( − 1 , 2 ) and ( − 1 , − 3 ) : d = 5 .
Calculate the distance between P ( − 1 , 2 ) and ( 4 , 2 ) : d = 5 .
Identify the three points that are 5 units away from P ( − 1 , 2 ) .
The three points 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 three points from the list that 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
Calculating Distances
Distance between P ( − 1 , 2 ) and ( 2 , − 2 ) :
d = ( 2 − ( − 1 ) ) 2 + ( − 2 − 2 ) 2 = ( 3 ) 2 + ( − 4 ) 2 = 9 + 16 = 25 = 5
Distance between P ( − 1 , 2 ) and ( − 6 , 7 ) :
d = ( − 6 − ( − 1 ) ) 2 + ( 7 − 2 ) 2 = ( − 5 ) 2 + ( 5 ) 2 = 25 + 25 = 50 = 5 2 ≈ 7.07
Distance between P ( − 1 , 2 ) and ( 1 , 5 ) :
d = ( 1 − ( − 1 ) ) 2 + ( 5 − 2 ) 2 = ( 2 ) 2 + ( 3 ) 2 = 4 + 9 = 13 ≈ 3.61
Distance between P ( − 1 , 2 ) and ( − 1 , − 3 ) :
d = ( − 1 − ( − 1 ) ) 2 + ( − 3 − 2 ) 2 = ( 0 ) 2 + ( − 5 ) 2 = 0 + 25 = 25 = 5
Distance between P ( − 1 , 2 ) and ( 4 , 2 ) :
d = ( 4 − ( − 1 ) ) 2 + ( 2 − 2 ) 2 = ( 5 ) 2 + ( 0 ) 2 = 25 + 0 = 25 = 5
Identifying Points at Distance 5 The points that are 5 units away from P ( − 1 , 2 ) are ( 2 , − 2 ) , ( − 1 , − 3 ) , and ( 4 , 2 ) .
Final Answer The three ordered pairs that describe points 5 units away from P ( − 1 , 2 ) are ( 2 , − 2 ) , ( − 1 , − 3 ) , and ( 4 , 2 ) .
Examples
Understanding distance between points is crucial in navigation and mapping. For instance, a GPS device calculates your distance from various cell towers to pinpoint your location. Similarly, in urban planning, calculating distances between residential areas, commercial centers, and public transportation hubs helps optimize city layouts for convenience and efficiency.
