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------------------------------------------------ For the given polar equation, write an equivalent rectangular equation.

r = 13

x² + y² = 169

(x + y)² = 169

x + y = 13

x² + y² = 169

In polar coordinates, [tex]r[/tex] represents the distance from the origin to a point [tex](r, \theta)[/tex] in the plane. In rectangular coordinates, [tex]r[/tex] can be expressed as:

[tex]r = \sqrt{x^2 + y^2}[/tex]

To convert this polar equation into a rectangular one, we substitute [tex]r[/tex] with [tex]\sqrt{x^2 + y^2}[/tex]:

[tex]\sqrt{x^2 + y^2} = 13[/tex]

To eliminate the square root, square both sides of the equation:

[tex]x^2 + y^2 = 169[/tex]

This is the equation of a circle with a center at the origin (0, 0) and a radius of 13 in the rectangular (Cartesian) coordinate system.

Let's summarize:

The original polar equation is [tex]r = 13[/tex].
The equivalent rectangular equation, after substituting and simplifying, is [tex]x^2 + y^2 = 169[/tex].

Thus, the rectangular equation represents a circle centered at the origin with a radius of 13 units.