To determine which expression gives the distance between the points [tex]\( (2,5) \)[/tex] and [tex]\( (-4,8) \)[/tex], let's analyze the given problem step by step.
The distance [tex]\( d \)[/tex] between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] in the coordinate plane is given by the distance formula:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Given the points [tex]\( (2,5) \)[/tex] and [tex]\( (-4,8) \)[/tex], we label these as [tex]\( (x_1, y_1) = (2, 5) \)[/tex] and [tex]\( (x_2, y_2) = (-4,8) \)[/tex].
Let's calculate each part:
1. Calculate [tex]\( x_2 - x_1 \)[/tex]:
[tex]\[ x_2 - x_1 = -4 - 2 = -6 \][/tex]
2. Calculate [tex]\( y_2 - y_1 \)[/tex]:
[tex]\[ y_2 - y_1 = 8 - 5 = 3 \][/tex]
3. Square both differences:
[tex]\[ (x_2 - x_1)^2 = (-6)^2 = 36 \][/tex]
[tex]\[ (y_2 - y_1)^2 = 3^2 = 9 \][/tex]
4. Add the squares:
[tex]\[ (x_2 - x_1)^2 + (y_2 - y_1)^2 = 36 + 9 = 45 \][/tex]
5. Take the square root to find the distance:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{45} \][/tex]
[tex]\[ d = \sqrt{(2 - (-4))^2 + (5 - 8)^2} \][/tex]
Now, let's look at the options and identify the correct one. We see:
A. [tex]\( (2+4)^2 + (5-8)^2 = 6^2 + (-3)^2 = 36 + 9 = 45 \)[/tex] (Not under a square root)
B. [tex]\( (2-4)^2 + (5-8)^2 = (-2)^2 + (-3)^2 = 4 + 9 = 13 \)[/tex] (Not matching the calculated squares)
C. [tex]\( \sqrt{(2+4)^2 + (5-8)^2} = \sqrt{6^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45} \)[/tex] (Distance matches)
D. [tex]\( \sqrt{(2-4)^2 + (5-8)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \)[/tex] (Distance does not match)
The correct expression that matches the distance between the points [tex]\( (2,5) \)[/tex] and [tex]\( (-4,8) \)[/tex] is:
[tex]\[ \boxed{\sqrt{(2+4)^2 + (5-8)^2}} \][/tex]
So the correct answer is:
[tex]\[ \boxed{C} \][/tex]
