To convert rectangular coordinates [tex]\((-5, 0)\)[/tex] to polar coordinates [tex]\((r, \theta)\)[/tex], we need to determine two things: the radius [tex]\(r\)[/tex] and the angle [tex]\(\theta\)[/tex].
1. Calculate the Radius [tex]\(r\)[/tex]:
The radius [tex]\(r\)[/tex] is the distance from the origin to the point [tex]\((-5, 0)\)[/tex]. We can find this using the Pythagorean theorem:
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
Given [tex]\(x = -5\)[/tex] and [tex]\(y = 0\)[/tex], we have:
[tex]\[
r = \sqrt{(-5)^2 + 0^2} = \sqrt{25} = 5
\][/tex]
2. Calculate the Angle [tex]\(\theta\)[/tex]:
The angle [tex]\(\theta\)[/tex] is measured from the positive x-axis to the point [tex]\((-5, 0)\)[/tex]. For a point where [tex]\(x < 0\)[/tex] and [tex]\(y = 0\)[/tex]:
[tex]\[
\theta = 180^\circ
\][/tex]
This is because the point lies on the negative x-axis, which corresponds to an angle of [tex]\(180^\circ\)[/tex].
3. Conclusion:
The polar coordinates that correspond to the rectangular coordinates [tex]\((-5, 0)\)[/tex] are:
[tex]\[
(r, \theta) = (5, 180^\circ)
\][/tex]
Thus, the correct set of polar coordinates matching the rectangular coordinates [tex]\((-5, 0)\)[/tex] is option B.
So the answer is:
[tex]\[
\boxed{\left(5, 180^\circ\right)}
\][/tex]
