Which expression gives the distance between the points $(2,5)$ and $(-4,8)$?
A. $(2-4)^2+(5-8)^2$
B. $\sqrt{(2+4)^2+(5-8)^2}$
C. $\sqrt{(2-4)^2+(5-8)^2}$
D. $(2+4)^2+(5-8)^2$
Answer (1)
Recall the distance formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
Substitute the given points ( 2 , 5 ) and ( − 4 , 8 ) into the formula: d = ( − 4 − 2 ) 2 + ( 8 − 5 ) 2 .
Simplify the expression: d = ( − 6 ) 2 + ( 3 ) 2 = ( 2 − ( − 4 ) ) 2 + ( 5 − 8 ) 2 .
Identify the correct option that matches the simplified expression: ( 2 − 4 ) 2 + ( 5 − 8 ) 2 .
Explanation
Problem Analysis We are given two points, ( 2 , 5 ) and ( − 4 , 8 ) , and we need to find the expression that represents the distance between them.
Distance Formula The distance formula between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) in a coordinate plane is given by:
d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
We will use this formula to calculate the distance between the given points.
Applying the Formula Let ( x 1 , y 1 ) = ( 2 , 5 ) and ( x 2 , y 2 ) = ( − 4 , 8 ) . Substituting these values into the distance formula, we get:
d = ( − 4 − 2 ) 2 + ( 8 − 5 ) 2
d = ( − 6 ) 2 + ( 3 ) 2
d = ( 2 − ( − 4 ) ) 2 + ( 5 − 8 ) 2
Now, we compare this expression with the given options.
Comparison with Options Comparing the calculated expression with the given options:
A. ( 2 − 4 ) 2 + ( 5 − 8 ) 2 - This is missing the square root and has incorrect subtraction in the first term. B. ( 2 + 4 ) 2 + ( 5 − 8 ) 2 - This has incorrect addition in the first term. C. ( 2 − 4 ) 2 + ( 5 − 8 ) 2 - This is the correct expression, as ( 2 − ( − 4 )) is not the same as ( 2 − 4 ) .
D. ( 2 + 4 ) 2 + ( 5 − 8 ) 2 - This is missing the square root and has incorrect addition in the first term.
Therefore, the correct expression is option C.
Final Answer The expression that gives the distance between the points ( 2 , 5 ) and ( − 4 , 8 ) is:
( 2 − ( − 4 ) ) 2 + ( 5 − 8 ) 2
Which simplifies to:
( 2 + 4 ) 2 + ( 5 − 8 ) 2
However, the correct option is C, which is ( 2 − ( − 4 ) ) 2 + ( 5 − 8 ) 2 = ( 2 + 4 ) 2 + ( 5 − 8 ) 2
Examples
The distance formula is a fundamental concept in coordinate geometry and has numerous real-world applications. For instance, it can be used in navigation systems to calculate the shortest distance between two locations, in computer graphics to determine distances between objects in a virtual environment, and in physics to calculate the magnitude of displacement vectors. Imagine you're using a GPS app to find the distance between your current location and a nearby coffee shop. The GPS uses coordinates and the distance formula to provide you with that information, helping you decide whether to walk or drive.
