What is the exact distance between the points $(4,-2)$ and $(-6,4)$?
Select the correct answer.
$4 \sqrt{10}$
$10 \sqrt{6}$
$2 \sqrt{34}$
$2 \sqrt{26}$
Answer (2)
Use the distance formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
Substitute the coordinates of the points: d = ( − 6 − 4 ) 2 + ( 4 − ( − 2 ) ) 2 .
Simplify the expression: d = 100 + 36 = 136 .
Simplify the square root: d = 2 34 .
The exact distance between the points is 2 34 .
Explanation
Problem Analysis We are given two points ( 4 , − 2 ) and ( − 6 , 4 ) , and we want to find the exact distance between them. We can 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.
Applying the Formula Let ( x 1 , y 1 ) = ( 4 , − 2 ) and ( x 2 , y 2 ) = ( − 6 , 4 ) . Substituting these values into the distance formula, we get: d = ( − 6 − 4 ) 2 + ( 4 − ( − 2 ) ) 2 d = ( − 10 ) 2 + ( 6 ) 2 d = 100 + 36 d = 136
Simplifying the Result Now, we simplify the square root: d = 4 × 34 d = 4 × 34 d = 2 34
Final Answer Therefore, the exact distance between the points ( 4 , − 2 ) and ( − 6 , 4 ) is 2 34 .
Examples
The distance formula is a fundamental concept in coordinate geometry and has numerous real-world applications. For example, imagine you are using a GPS to navigate. The GPS uses coordinates to pinpoint your location and the location of your destination. The distance formula is then used to calculate the shortest distance between these two points, providing you with an estimated travel time and directions. This principle is also used in mapping, surveying, and various fields of engineering to measure distances accurately.
The distance between the points (4, -2) and (-6, 4) is found using the distance formula, resulting in the final answer of 2√34. This is derived by substituting the coordinates into the formula and simplifying the expression. Thus, the correct answer is 2√34.
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