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A. L′(−5,−3),H′(−3,2),E′(−1,0)𝐿′(−5,−3),𝐻′(−3,2),𝐸′(−1,0)

B. L′(0,−3),H′(2,2),E′(4,0)𝐿′(0,−3),𝐻′(2,2),𝐸′(4,0)

C. L′(−1,−5),H′(1,0),E′(3,−2)𝐿′(−1,−5),𝐻′(1,0),𝐸′(3,−2)

D. L′(−2,0),H′(0,5),E′(2,3)


To determine the new coordinates of the vertices of the figure after the given transformation, we need to apply the transformation \( T_{<-1,1>} \), which indicates a translation by \(-1\) units in the \( x \)-direction and \( 1 \) unit in the \( y \)-direction.

Let's apply this transformation to each vertex of the given figure.

### Original Coordinates:
- \( L = (-1, -4) \)
- \( H = (1, -1) \)
- \( E = (3, -3) \)

### Applying the Transformation \( T_{<-1,1>} \):
To find the new coordinates \( (x', y') \) of each vertex after the translation, use the following formulas:
- \( x' = x - 1 \)
- \( y' = y + 1 \)

#### For vertex \( L \):
- Original coordinates: \( (-1, -4) \)
- New coordinates: \( (-1 - 1, -4 + 1) = (-2, -3) \)

#### For vertex \( H \):
- Original coordinates: \( (1, -1) \)
- New coordinates: \( (1 - 1, -1 + 1) = (0, 0) \)

#### For vertex \( E \):
- Original coordinates: \( (3, -3) \)
- New coordinates: \( (3 - 1, -3 + 1) = (2, -2) \)

### New Coordinates:
- \( L' = (-2, -3) \)
- \( H' = (0, 0) \)
- \( E' = (2, -2) \)

Let's match these results with the given choices:

A. \( L'(-5, -3), H'(-3, 2), E'(-1, 0) \)  
B. \( L'(0, -3), H'(2, 2), E'(4, 0) \)  
C. \( L'(-1, -5), H'(1, 0), E'(3, -2) \)  
D. \( L'(-2, 0), H'(0, 5), E'(2, 3) \)

None of the provided choices match the coordinates obtained from the transformation. It seems there may have been an error in the interpretation of the coordinates or the problem statement. Please check the problem statement or provided choices again.

Would you like more details or have any questions?

Here are 8 related questions to expand on this topic:

1. What is the definition of a geometric transformation?
2. How does a translation affect the coordinates of a point?
3. What are other types of transformations in geometry?
4. How do you represent a rotation transformation on a coordinate plane?
5. What is the effect of a reflection transformation on the coordinates of a point?
6. How can you compose multiple transformations?
7. What is the general formula for translating a point by a vector \( \langle a, b \rangle \)?
8. How would you apply a scaling transformation to these coordinates?

**Tip:** Always double-check the coordinates and the transformation rules to ensure accurate calculations.

Explore how to apply coordinate transformations to solve a geometric problem involving translations. Understand the effects of translations on geometric figures and verify transformations against provided options. Suitable for high school students.

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