Find the distance between the points: $(-8,2)$ and $(4,-8)$
Answer (2)
Use the distance formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
Substitute the given points ( − 8 , 2 ) and ( 4 , − 8 ) into the formula.
Simplify the expression: d = ( 4 − ( − 8 ) ) 2 + ( − 8 − 2 ) 2 = 1 2 2 + ( − 10 ) 2 = 144 + 100 = 244 .
The distance between the points is 244 .
Explanation
Problem Analysis We are given two points, ( − 8 , 2 ) and ( 4 , − 8 ) , and we want to find the distance between them. We can use the distance formula to find the distance.
Distance Formula The distance formula is given by: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the two points.
Substitute Values In this case, ( x 1 , y 1 ) = ( − 8 , 2 ) and ( x 2 , y 2 ) = ( 4 , − 8 ) . Substituting these values into the distance formula, we get: d = ( 4 − ( − 8 ) ) 2 + ( − 8 − 2 ) 2 d = ( 4 + 8 ) 2 + ( − 10 ) 2 d = ( 12 ) 2 + ( − 10 ) 2 d = 144 + 100 d = 244
Simplify the Expression We can simplify 244 by factoring 244. 244 = 4 × 61 , so 244 = 4 × 61 = 4 × 61 = 2 61 .
The distance between the points ( − 8 , 2 ) and ( 4 , − 8 ) is 244 or 2 61 .
Final Answer The distance between the points ( − 8 , 2 ) and ( 4 , − 8 ) is 244 ≈ 15.62 . The question asks for d 2 , which is 244.
Examples
The distance formula is a fundamental concept in coordinate geometry and has numerous real-world applications. For instance, civil engineers use it to calculate distances between points on a map when planning infrastructure projects like roads or bridges. Similarly, in navigation, the distance formula helps determine the shortest path between two locations, which is crucial for efficient route planning in aviation and maritime transport. Even in everyday life, we use the concept of distance when estimating travel times or determining the proximity of different places.
The distance between the points ( − 8 , 2 ) and ( 4 , − 8 ) , calculated using the distance formula, is 244 or approximately 2 61 , which is about 15.62 units.
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