To find the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] on the Cartesian plane, we use the distance formula:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Let's use the coordinates of the two points given:
[tex]\[
(x_1, y_1) = (1, 4)
\][/tex]
[tex]\[
(x_2, y_2) = (-10, -3)
\][/tex]
First, we find the differences in the x-coordinates and y-coordinates:
[tex]\[
x_2 - x_1 = -10 - 1 = -11
\][/tex]
[tex]\[
y_2 - y_1 = -3 - 4 = -7
\][/tex]
Next, we square these differences:
[tex]\[
(-11)^2 = 121
\][/tex]
[tex]\[
(-7)^2 = 49
\][/tex]
We then add these squared values together:
[tex]\[
121 + 49 = 170
\][/tex]
Finally, we take the square root of this sum to find the distance:
[tex]\[
d = \sqrt{170}
\][/tex]
Therefore, the distance between the points [tex]\((1, 4)\)[/tex] and [tex]\((-10, -3)\)[/tex] can be expressed as a simplified radical:
[tex]\[
\boxed{\sqrt{170}}
\][/tex]
