Question

An engineer is planning to construct triangular halls on a new construction site. A scale model of the engineer's triangular halls is shown below. Each unit represents 10 feet.

Hall A has vertices at (-4, 0), (-2, 2), (-2, -2), Hall B has vertices at (4, 0), (2, -2), (2, 2), Hall C has its vertices at (5, -5), (10, 0), (5, 5).

Which of the following statements are true about this scale model? Select three statements that apply.
A.
Hall C is similar to Hall A.

B.
Hall C is similar to Hall B.

C.
Hall C is congruent to Hall A.

D.
Hall B is congruent to Hall A.

E.
The scale factor of dilation applied to Hall B to produce Hall C is 5.

Answer 1

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.

1. **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

2. **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

3. **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.