To determine the relationships and similarities between the triangles representing the halls, we need to analyze the given vertices and their respective lengths and angles.
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Hall A vertices:
- A: (-4, 0)
- B: (-2, 2)
- C: (-2, -2)
Calculating the side lengths using the distance formula:
- AB = √[(-2 + 4)² + (2 - 0)²] = √(2² + 2²) = √8 = 2√2
- AC = √[(-2 + 4)² + (-2 - 0)²] = √(2² + 2²) = √8 = 2√2
- BC = √[(-2 + 2)² + (2 + 2)²] = √(0² + 4²) = 4
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Hall B vertices:
- A: (4, 0)
- B: (2, -2)
- C: (2, 2)
Following the same calculations for lengths:
- AB = √[(2 - 4)² + (-2 - 0)²] = √(-2)² + (-2)² = √8 = 2√2
- AC = √[(2 - 4)² + (2 - 0)²] = √(-2)² + 2² = √8 = 2√2
- BC = √[(2 - 2)² + (2 + 2)²] = √(0² + 4²) = 4
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Hall C vertices:
- A: (5, -5)
- B: (10, 0)
- C: (5, 5)
Again, compute the side lengths:
- AB = √[(10 - 5)² + (0 + 5)²] = √(5² + 5²) = √50 = 5√2
- AC = √[(5 - 5)² + (5 + 5)²] = √(0 + 10²) = 10
- BC = √[(10 - 5)² + (5 - 0)²] = √(5² + 5²) = √50 = 5√2
Now we can evaluate each option:
A. Hall C is similar to Hall A.
- Not true. The side lengths are not proportional; Hall C has sides of \( (5√2, 10, 5√2) \) and Hall A has sides \( (2√2, 2√2, 4) \).
B. Hall C is similar to Hall B.
- True. The ratios of their corresponding sides give a consistent ratio of 5:2 (for example, \( \frac{5√2}{2√2} = 2.5 \) and \( \frac{10}{4} = 2.5 \)).
C. Hall C is congruent to Hall A.
- Not true. The triangles are not the same size as they do not have equal lengths.
D. Hall B is congruent to Hall A.
- True. They have the same side lengths: \( 2√2, 2√2, 4 \).
E. The scale factor of dilation applied to Hall B to produce Hall C is 5.
- True. Appearance suggests that Hall C is a scaled up version of Hall B by a factor of \( 5/2 \), however more directly, looking at the side lengths confirms \( 5:2 \) for sides.
The three true statements are:
- B. Hall C is similar to Hall B.
- D. Hall B is congruent to Hall A.
- E. The scale factor of dilation applied to Hall B to produce Hall C is 5.
Thus, the selected statements are B, D, and E.
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