To determine which expression gives the distance between the points [tex]\((5,1)\)[/tex] and [tex]\((9,-6)\)[/tex], we need to use the distance formula for two points in a Cartesian plane. The distance [tex]\( d \)[/tex] between points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For the points [tex]\((5, 1)\)[/tex] and [tex]\((9, -6)\)[/tex]:
1. Calculate the difference in the x-coordinates:
[tex]\[x_2 - x_1 = 9 - 5 = 4\][/tex]
2. Calculate the difference in the y-coordinates:
[tex]\[y_2 - y_1 = -6 - 1 = -7\][/tex]
3. Square each of these differences:
[tex]\[(4)^2 = 16\][/tex]
[tex]\[(-7)^2 = 49\][/tex]
4. Add the squares of the differences:
[tex]\[16 + 49 = 65\][/tex]
5. Take the square root of the sum:
[tex]\[\sqrt{65} \approx 8.06\][/tex]
Based on the steps above, we can see that the distance between the points [tex]\((5,1)\)[/tex] and [tex]\((9,-6)\)[/tex] is given by:
[tex]\[ \sqrt{(9 - 5)^2 + (-6 - 1)^2} = \sqrt{4^2 + (-7)^2} \][/tex]
Comparing this to the options provided:
A. [tex]\((5-9)^2+(1-6)^2\)[/tex]
B. [tex]\(\sqrt{(5-9)^2+(1+6)^2}\)[/tex]
C. [tex]\(\sqrt{(5-9)^2+(1-6)^2}\)[/tex]
D. [tex]\((5-9)^2+(1+6)^2\)[/tex]
We can simplify and match the correct expression:
- Option A does not include the square root, so it cannot be the distance formula.
- Option B has an incorrect sign in the y-coordinate difference.
- Option C correctly matches our derived formula.
- Option D does not include the square root and also has an incorrect sign in the y-coordinate difference.
Therefore, the correct expression is:
[tex]\[ \boxed{\text{C.}} \sqrt{(5-9)^2+(1-6)^2} \][/tex]
