Question

do the following side lengths create a triangle? 5, 10, 15 sometimes no no clue yes

Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that for three side lengths \(a\), \(b\), and \(c\) to form a triangle, the sum of any two sides must be greater than the third side. Mathematically, this means:

• \(a + b > c\)
• \(a + c > b\)
• \(b + c > a\)

Step2: Apply the theorem to the given side lengths

Given the side lengths \(5\), \(10\), and \(15\), let's check each inequality:

• Check \(5 + 10 > 15\): \(5 + 10 = 15\), and \(15\) is not greater than \(15\) (it's equal).
• Check \(5 + 15 > 10\): \(5 + 15 = 20\), and \(20 > 10\) (this holds).
• Check \(10 + 15 > 5\): \(10 + 15 = 25\), and \(25 > 5\) (this holds).

Since one of the inequalities (\(5 + 10 > 15\)) does not hold (it's equal, not greater), the side lengths do not satisfy the triangle inequality theorem.

Answer:

The side lengths \(5\), \(10\), and \(15\) do not create a triangle, so the correct option is the blue "No" button (the one with the diamond symbol). In terms of the options provided:

• The blue "No" button (with the diamond) is the correct choice. So the answer is the option labeled "No" (the blue one with the diamond). If we consider the options as:
• Red: Sometimes
• Blue: No
• Yellow: No Clue
• Green: Yes

Then the correct answer is the blue "No" option, so:

Blue. No (the one with the diamond symbol)