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 values represent possible lengths between amy and becky so that amy, becky, and claire are standing at the vertices of a triangle? options: 4 ft, 24 ft, 20 ft, 12 ft, 15 ft, 9 ft

Explanation:

Step1: Recall triangle inequality theorem

For a triangle with side lengths \(a\), \(b\), \(c\), the triangle inequality states that \(|a - b| < c < a + b\). Here, let \(a = 15\) (distance from Claire to Becky), \(b = 6\) (distance from Claire to Amy), and \(c\) be the distance from Amy to Becky. So we need to find \(c\) such that \(|15 - 6| < c < 15 + 6\), i.e., \(9 < c < 21\).

Step2: Check each option

• \(4\) ft: \(4<9\), not valid.
• \(9\) ft: \(9 = 9\), not valid (needs to be greater than \(9\)).
• \(12\) ft: \(9<12<21\), valid.
• \(15\) ft: \(9<15<21\), valid.
• \(20\) ft: \(9<20<21\), valid.
• \(24\) ft: \(24>21\), not valid.

Answer:

12 ft, 15 ft, 20 ft (the corresponding checkboxes should be selected)