https://file.aiquickdraw.com/mathbot/file/1727210675750npsxu0nq.jpg

The image depicts a rhombus inscribed in a coordinate plane, with its vertices labeled \( R(0, b) \), \( U(-a, 0) \), and the remaining two vertices \( S \) and \( T \) having unknown coordinates. The task is to find the coordinates of points \( S \) and \( T \).

### Key Observations:
1. **Symmetry**: The rhombus is symmetric with respect to both the x-axis and the y-axis, which suggests that the coordinates of points \( S \) and \( T \) can be inferred using symmetry.
   
2. **Vertices of a Rhombus**: The vertices of the rhombus lie on the axes or near the axes:
   - \( U(-a, 0) \) is on the negative x-axis.
   - \( R(0, b) \) is on the positive y-axis.
   
   By symmetry:
   - \( S \) should be on the positive x-axis and have the same y-coordinate as \( U \) (which is 0).
   - \( T \) should be on the negative y-axis and have the same x-coordinate as \( R \) (which is 0).

### Finding Coordinates:
- **Point \( S \)**: Since it lies on the positive x-axis and mirrors \( U(-a, 0) \), the x-coordinate of \( S \) must be \( a \), and the y-coordinate is \( 0 \).
  Therefore, the coordinates of \( S \) are \( S(a, 0) \).
  
- **Point \( T \)**: As it mirrors \( R(0, b) \) on the negative y-axis, the x-coordinate remains \( 0 \), and the y-coordinate is \( -b \).
  Therefore, the coordinates of \( T \) are \( T(0, -b) \).

### Final Answer:
- \( S(a, 0) \)
- \( T(0, -b) \)

Would you like more details or clarifications on how I arrived at these coordinates?

Here are five related questions you might want to explore:
1. How can we derive the area of the rhombus based on the coordinates of its vertices?
2. What is the general equation of a rhombus in the coordinate plane?
3. How do we determine the lengths of the diagonals of this rhombus using the given points?
4. Can we apply the distance formula to confirm that all sides of the rhombus are equal?
5. How can the slope of the diagonals help us verify the symmetry of the rhombus?

**Tip**: When solving geometry problems in the coordinate plane, symmetry often helps simplify calculations by reducing the number of unknowns.

A rhombus is inscribed in a coordinate plane with vertices labeled R(0, b), U(-a, 0), and two unknown points S and T. Find the coordinates of S and T.

Learn how to find the coordinates of the vertices of a rhombus inscribed in a coordinate plane by using symmetry. This problem covers key geometry concepts such as symmetry, properties of a rhombus, and coordinate geometry. Suitable for students in Grades 9-10.

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