To determine the distance between point [tex]\(A(2,6)\)[/tex] and point [tex]\(B(5,1)\)[/tex], we use the distance formula. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
For Silvia’s problem, we identify the coordinates of the two points:
- Point [tex]\(A\)[/tex] has coordinates [tex]\((x_1, y_1) = (2, 6)\)[/tex]
- Point [tex]\(B\)[/tex] has coordinates [tex]\((x_2, y_2) = (5, 1)\)[/tex]
Substituting these values into the distance formula, we get:
[tex]\[
\text{Distance} = \sqrt{(5 - 2)^2 + (1 - 6)^2}
\][/tex]
Now let's verify each of the given expressions to check which one correctly matches our derived formula:
1. [tex]\(\sqrt{(5-2)^2+(1-6)^2}\)[/tex]
- This matches the formula used above.
2. [tex]\(\sqrt{(5-1)^2+(2-6)^2}\)[/tex]
- Using this formula would mean switching the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates incorrectly, which does not correspond to the correct application of the distance formula.
3. [tex]\(\sqrt{(6-5)^2+(2-1)^2}\)[/tex]
- This uses incorrect differences for both coordinate pairs of the points.
4. [tex]\(\sqrt{(5-6)^2+(1-2)^2}\)[/tex]
- While the differences are used in reversed sign, the order in subtraction also does not match the required point coordinates directly.
Thus, the correct expression Silvia should use to calculate the distance between point [tex]\(A(2,6)\)[/tex] and point [tex]\(B(5,1)\)[/tex] is:
[tex]\[
\sqrt{(5-2)^2+(1-6)^2}
\][/tex]
