Question

A triangle upper A upper B upper C. The edge upper A upper B is labeled c. The edge upper A upper C is labeled b. The edge upper B upper C is labeled a.
(Diagram is not to scale.)

In the diagram, Amanda lives at A, Chad lives at C, and B represents the baseball diamond. Amanda tells Chad she walks 2,200 ft. to the baseball diamond. Chad says he only walks 1,320 ft., so she should come to his house and walk with him. What is a possible distance from Amanda’s (A) to Chad’s (C)?

(1 point)
Responses

880 ft.
880 ft.

1,040 ft.
1,040 ft.

3,520 ft.
3,520 ft.

4,400 ft.
4,400 ft.
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Answer 1

In this situation, we need to use the triangle inequality theorem, which states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following inequalities must hold:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

In this case:

- \(AB = c = 2200\) ft (distance Amanda walks to the baseball diamond)
- \(AC = b = d\) (distance from Amanda's house to Chad's house, which we need to determine)
- \(BC = a = 1320\) ft (distance Chad walks to the baseball diamond)

Using the triangle inequality, we need to examine the example of \(AC\):

1. \(AB + AC > BC\)
\[
2200 + d > 1320 \implies d > 1320 - 2200 \implies d > -880 \quad \text{(always true)}
\]

2. \(AB + BC > AC\)
\[
2200 + 1320 > d \implies 3520 > d \implies d < 3520
\]

3. \(AC + BC > AB\)
\[
d + 1320 > 2200 \implies d > 2200 - 1320 \implies d > 880
\]

From these three inequalities, we find the following constraints for \(d\):

\[
880 < d < 3520
\]

Thus, \(d\) can be any distance greater than 880 ft and less than 3520 ft.

From the answer choices:
- 880 ft (not valid, since \(d\) must be greater than 880)
- 1,040 ft (valid)
- 3,520 ft (not valid, since \(d\) must be less than 3520)
- 4,400 ft (not valid)

The only possible distance is:

**1,040 ft.**