Barry is trying to calculate the distance between point E(3, 1) and point F(4, 7). Which of the following expressions will he use?
A. √(7-1)²+(4-3)²
B. √(7-3)²+(4-1)²
C. (7-4)²+(3-1)²
D. √(4-7)²+(1-3)²
Answer (2)
The distance between two points E(3, 1) and F(4, 7) is calculated using the distance formula. The distance formula is d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . Substituting the coordinates, we get d = ( 4 − 3 ) 2 + ( 7 − 1 ) 2 .
Recall the distance formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
Substitute the coordinates of E(3, 1) and F(4, 7) into the formula: d = ( 4 − 3 ) 2 + ( 7 − 1 ) 2 .
Simplify the expression: d = ( 1 ) 2 + ( 6 ) 2 = 1 + 36 = 37 .
The expression Barry will use is: ( 7 − 1 ) 2 + ( 4 − 3 ) 2 .
Explanation
Understanding the problem The problem asks for the expression to calculate the distance between two points E(3, 1) and F(4, 7). We need to use the distance formula.
Stating the distance formula 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 In this case, E ( 3 , 1 ) is ( x 1 , y 1 ) and F ( 4 , 7 ) is ( x 2 , y 2 ) .
Applying the formula to the given points Substitute the coordinates of points E and F into the distance formula: d = ( 4 − 3 ) 2 + ( 7 − 1 ) 2 This simplifies to: d = ( 1 ) 2 + ( 6 ) 2 d = 1 + 36 d = 37
Comparing with the given options Now we compare the derived expression ( 4 − 3 ) 2 + ( 7 − 1 ) 2 with the given options to find the correct one. The expression ( 4 − 3 ) 2 + ( 7 − 1 ) 2 matches the option ( 7 − 1 ) 2 + ( 4 − 3 ) 2 . Note that ( 4 − 3 ) 2 = ( 3 − 4 ) 2 and ( 7 − 1 ) 2 = ( 1 − 7 ) 2 , so ( 4 − 3 ) 2 + ( 7 − 1 ) 2 = ( 7 − 1 ) 2 + ( 4 − 3 ) 2 = ( 4 − 7 ) 2 + ( 1 − 3 ) 2 .
Final Answer The correct expression to calculate the distance between point E(3, 1) and point F(4, 7) is ( 7 − 1 ) 2 + ( 4 − 3 ) 2 .
Examples
The distance formula is a fundamental concept in geometry and has many real-world applications. For example, it can be used in navigation to calculate the shortest distance between two locations on a map. In sports, it can be used to track the distance a player runs during a game. In computer graphics, it's used to determine distances between objects in a virtual environment. Suppose you want to find the distance between your home and a nearby park. If you know the coordinates of both locations on a map, you can use the distance formula to calculate the straight-line distance between them. This can help you estimate the time it will take to walk or bike to the park.
To calculate the distance between points E(3, 1) and F(4, 7), we use the distance formula, resulting in the expression ( 4 − 3 ) 2 + ( 7 − 1 ) 2 . The correct choice from the options given is A: ( 7 − 1 ) 2 + ( 4 − 3 ) 2 .
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