Find the distance between the points $(9,-6)$ and $(-4,7)$.
A) $13 \sqrt{2}$
B) $3 \sqrt{39}$
C) $2 \sqrt{85}$
D) $3 \sqrt{39}$
Answer (2)
Apply the distance formula: ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
Substitute the coordinates: ( − 4 − 9 ) 2 + ( 7 − ( − 6 ) ) 2 .
Simplify the expression: ( − 13 ) 2 + ( 13 ) 2 = 169 + 169 = 338 .
Obtain the final answer: 13 2 .
Explanation
Problem Analysis Let's find the distance between the points ( 9 , − 6 ) and ( − 4 , 7 ) . We'll use the distance formula to solve this problem.
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. In our case, ( x 1 , y 1 ) = ( 9 , − 6 ) and ( x 2 , y 2 ) = ( − 4 , 7 ) .
Calculations Substitute the given coordinates into the distance formula:
d = ( − 4 − 9 ) 2 + ( 7 − ( − 6 ) ) 2
Simplify the expression:
d = ( − 13 ) 2 + ( 13 ) 2
Calculate the squares:
d = 169 + 169
Combine the terms:
d = 338
Simplify the square root:
d = 169 ⋅ 2 = 13 2
Final Answer The distance between the points ( 9 , − 6 ) and ( − 4 , 7 ) is 13 2 .
Examples
The distance formula is a fundamental concept in coordinate geometry and has many practical applications. For example, civil engineers use the distance formula to calculate the lengths of roads or bridges on a coordinate plane. Similarly, in navigation, sailors or pilots use coordinate systems and the distance formula to determine the shortest distance between two locations. In computer graphics and game development, the distance formula is used to calculate distances between objects, which is essential for collision detection and other interactive features.
The distance between the points (9, -6) and (-4, 7) is calculated using the distance formula, resulting in an answer of 13√2. Therefore, the correct choice is A) 13√2.
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