Question

can the sides of a triangle have lengths 3, 9, and 10? yes no a yes b no

Explanation:

Step1: Recall triangle inequality theorem

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

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

Step2: Apply the theorem to the given side lengths

Let \(a = 3\), \(b = 9\), and \(c = 10\).

• Check \(a + b>c\): \(3+9 = 12\), and \(12>10\), so this inequality holds.
• Check \(a + c>b\): \(3 + 10=13\), and \(13>9\), so this inequality holds.
• Check \(b + c>a\): \(9+10 = 19\), and \(19>3\), so this inequality holds.

Since all three triangle inequalities are satisfied, a triangle can have side lengths 3, 9, and 10.

Answer:

a. Yes