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The problem involves converting polar coordinates and identifying points that correspond to the same location but with different values of \( r \) and \( \theta \). Here is a breakdown of the information given:

- The point given is \( \left(5, \frac{\pi}{4}\right) \).
- To find the point directly opposite this point, we add \( \pi \) to the angle, resulting in the point \( \left(5, \frac{\pi}{4} + \pi\right) = \left(5, \frac{5\pi}{4}\right) \).
- The problem then introduces the concept that using \( -r \) (i.e., a negative radius) can also represent the same point if you adjust the angle accordingly.

### Solution to the Blank:
The statement mentions that a pair of polar coordinates \( (-r, \theta) \) is equivalent to \( (r, \theta + \pi) \). Therefore, if we want to represent the point \( \left(5, \frac{5\pi}{4}\right) \) with a negative radius, we take \( r = -5 \) and the angle remains \( \frac{\pi}{4} \).

Thus, the polar coordinates for the same point using a negative radius are:
\[
(-5, \frac{\pi}{4})
\]

Would you like more details or have any questions about this explanation?

### Related Questions:
1. How do you convert between polar and Cartesian coordinates?
2. What is the significance of adding \( \pi \) to an angle in polar coordinates?
3. How do negative values of \( r \) affect the representation of a point in polar coordinates?
4. What are some examples of polar coordinates that represent the same point using different values of \( r \) and \( \theta \)?
5. How does the conversion formula between polar and Cartesian coordinates change with negative \( r \) values?

### Tip:
In polar coordinates, adding \( 2\pi \) to any angle results in the same point, as it represents a full rotation around the origin. This property is useful when converting between different representations.

Find polar coordinates that represent the same point as (5, π/4) using a negative radius.

This problem demonstrates how to convert polar coordinates (5, π/4) to their equivalent form using a negative radius. Explore the transformation involving angle shifts and negative radii representation, along with key mathematical formulas. Suitable for high school students (Grades 10-12).

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