To find the distance between the points [tex]\((4, 6)\)[/tex] and [tex]\((7, -3)\)[/tex], we'll use the distance formula, which is derived from the Pythagorean theorem. The distance formula is:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Let's assign [tex]\((x_1, y_1) = (4, 6)\)[/tex] and [tex]\((x_2, y_2) = (7, -3)\)[/tex].
Now, substitute the coordinates into the formula:
1. Calculate [tex]\(x_2 - x_1\)[/tex]:
[tex]\[
x_2 - x_1 = 7 - 4 = 3
\][/tex]
2. Calculate [tex]\(y_2 - y_1\)[/tex]:
[tex]\[
y_2 - y_1 = -3 - 6 = -9
\][/tex]
3. Substitute [tex]\(3\)[/tex] and [tex]\(-9\)[/tex] into the formula and square them:
[tex]\[
(x_2 - x_1)^2 = 3^2 = 9
\][/tex]
[tex]\[
(y_2 - y_1)^2 = (-9)^2 = 81
\][/tex]
4. Add the squared differences:
[tex]\[
(x_2 - x_1)^2 + (y_2 - y_1)^2 = 9 + 81 = 90
\][/tex]
5. Take the square root of the sum to find the distance:
[tex]\[
\sqrt{90}
\][/tex]
This simplifies to approximately:
[tex]\[
\sqrt{90} \approx 9.486832980505138
\][/tex]
Now, let’s evaluate each given option to find the correct one:
A. [tex]\((4-7)^2 + (6-3)^2 = (-3)^2 + 3^2 = 9 + 9 = 18\)[/tex]
B. [tex]\(\sqrt{(4-7)^2 + (6-3)^2} = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18}\)[/tex]
C. [tex]\(\sqrt{(4-7)^2 + (6+3)^2} = \sqrt{(-3)^2 + 9^2} = \sqrt{9 + 81} = \sqrt{90}\)[/tex]
D. [tex]\((4-7)^2 + (6+3)^2 = (-3)^2 + 9^2 = 9 + 81 = 90\)[/tex]
From these evaluations, the correct expression that matches our derived distance formula is:
[tex]\[
\sqrt{(4-7)^2 + (6+3)^2}
\][/tex]
Therefore, the answer is C.
