What is the image point of left parenthesis, minus, 5, comma, minus, 7, right parenthesis(−5,−7) after the transformation R, start subscript, 180, degrees, end subscript, circle, T, start subscript, 0, comma, 0, end subscriptR 
180 
∘
 
​	
 ∘T 
0,0
​	
 ?

To find the image of the point \((-5, -7)\) after the transformation \( R_{180^\circ} \circ T_{0,0} \), we need to follow these steps:

1. **Rotation by \(180^\circ\)** about the origin:
   - When a point \((x, y)\) is rotated by \(180^\circ\) around the origin, the transformation results in the point \((-x, -y)\).
   
   So, applying the \(180^\circ\) rotation to the point \((-5, -7)\):
   \[
   (-5, -7) \xrightarrow{R_{180^\circ}} (5, 7)
   \]

Thus, the image of the point \((-5, -7)\) after the transformation \(R_{180^\circ} \circ T_{0,0}\) is **(5, 7)**.

Would you like more details or clarifications on this transformation? 

Here are 5 related questions to explore further:

1. How does the transformation change if we rotate by \(90^\circ\) instead of \(180^\circ\)?
2. What would happen if we applied a translation after the rotation?
3. How does rotating a point by \(180^\circ\) affect its distance from the origin?
4. Can you apply the same transformation to any point on the coordinate plane?
5. What is the difference between a rotation around the origin and around another point?

**Tip:** When rotating points, always pay attention to the center of rotation—it determines how the coordinates change!

What is the image point of (-5,-7) after the transformation R_180 degrees ∘ T_0,0?

Learn how to find the image of the point (-5, -7) after a 180-degree rotation around the origin. This detailed solution covers key concepts in coordinate geometry and transformations, suitable for students in Grades 8-10.

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