https://file.aiquickdraw.com/mathbot/file/1727261871159y4s0uqdr.jpg

To solve the given problem, let's address each part step-by-step.

### Problem Overview
1. The task is to determine the image (transformation) of the triangle \( ABC \) and the parallelogram \( PQRS \) given specific transformations:
   - Part (a): Transform triangle \( ABC \) by translation \( T(5, -3) \).
   - Part (b): Transform parallelogram \( PQRS \) by translation \( T(-6, 2) \).

### Solution

**(a) Transforming Triangle \( ABC \) with Translation \( T(5, -3) \)**

**Original coordinates of triangle \( ABC \):**
- \( A = (-5, 1) \)
- \( B = (-3, 0) \)
- \( C = (-2, 4) \)

**Translation vector:** \( T(5, -3) \) means moving each point 5 units to the right and 3 units down.

**New coordinates after transformation:**
- \( A' = (-5 + 5, 1 - 3) = (0, -2) \)
- \( B' = (-3 + 5, 0 - 3) = (2, -3) \)
- \( C' = (-2 + 5, 4 - 3) = (3, 1) \)

**(b) Transforming Parallelogram \( PQRS \) with Translation \( T(-6, 2) \)**

**Original coordinates of parallelogram \( PQRS \):**
- \( P = (2, -2) \)
- \( Q = (5, -1) \)
- \( R = (4, 1) \)
- \( S = (1, 0) \)

**Translation vector:** \( T(-6, 2) \) means moving each point 6 units to the left and 2 units up.

**New coordinates after transformation:**
- \( P' = (2 - 6, -2 + 2) = (-4, 0) \)
- \( Q' = (5 - 6, -1 + 2) = (-1, 1) \)
- \( R' = (4 - 6, 1 + 2) = (-2, 3) \)
- \( S' = (1 - 6, 0 + 2) = (-5, 2) \)

### Summary of the New Coordinates:
- **Triangle ABC after translation \( T(5, -3) \):** 
  - \( A' = (0, -2) \)
  - \( B' = (2, -3) \)
  - \( C' = (3, 1) \)
- **Parallelogram PQRS after translation \( T(-6, 2) \):** 
  - \( P' = (-4, 0) \)
  - \( Q' = (-1, 1) \)
  - \( R' = (-2, 3) \)
  - \( S' = (-5, 2) \)

Would you like further details or have any questions? 

Here are some related questions you might explore:
1. How would the transformation change if the vector was \( T(-5, 3) \)?
2. What are the properties of translation transformations?
3. How do you determine the translation vector from given original and image points?
4. Can you describe how to plot these new points on a coordinate plane?
5. How does translation differ from other transformations like rotation and reflection?

**Tip:** Always double-check the direction of your translation vector (positive for right/up, negative for left/down) to avoid mistakes!

Learn how to apply translation on triangle ABC and parallelogram PQRS. The solution involves translating shapes using vectors on a coordinate plane. Suitable for students in Grades 8-10.

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