Question

Select all the correct answers.

Claire, Amy, and Becky are at an art show. They are currently looking at different pieces of art. Claire is standing 6 feet from Amy and 15 feet from Becky. Which of the following three values represent possible lengths between Amy and Becky so that Amy, Becky, and Claire are standing at the vertices of a triangle?

A. 24 ft
B. 20 ft
C. 9 ft
D. 15 ft
E. 12 ft
F. 4 ft

Answer 1

To determine whether Amy, Becky, and Claire are standing at the vertices of a triangle, we need to use the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given:
- Claire to Amy: 6 feet
- Claire to Becky: 15 feet

Let's denote:
- \( a \) as the distance between Claire and Amy (6 feet)
- \( b \) as the distance between Claire and Becky (15 feet)
- \( c \) as the distance between Amy and Becky

A value \( c \) will form a triangle with \( a \) and \( b \) if:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Substitute \( a = 6 \) feet and \( b = 15 \) feet into these conditions:
1. \( 6 + 15 > c \) ⟹ \( c < 21 \)
2. \( 6 + c > 15 \) ⟹ \( c > 9 \)
3. \( 15 + c > 6 \) ⟹ (This condition is always true since \( c \) is positive and \( 15 > -c \)).

Therefore, \( c \) must satisfy \( 9 < c < 21 \).

Let's check which values among the given choices satisfy this range:
A. 24 ft (not in \( 9 < c < 21 \) range)
B. 20 ft (in \( 9 < c < 21 \) range)
C. 9 ft (not in \( 9 < c < 21 \) range)
D. 15 ft (in \( 9 < c < 21 \) range)
E. 12 ft (in \( 9 < c < 21 \) range)
F. 4 ft (not in \( 9 < c < 21 \) range)

Thus, the correct answers are:
B. 20 ft
D. 15 ft
E. 12 ft