Triangle Inequality Theorem Quick Check

Question

4 of 54 of 5 Items
Question
Apply the Triangle Inequality Theorem to determine whether sides with lengths of 5 inches, 10 inches, and 15 inches form a triangle.(1 point)
Responses

Yes, because the sum of 5+10 is less than or equal to 15.
Yes, because the sum of 5 plus 10 is less than or equal to 15.

No, because the sum of 5+10 is not greater than 15.
No, because the sum of 5 plus 10 is not greater than 15.

Yes, because the sum of 5+10 is not greater than 15.
Yes, because the sum of 5 plus 10 is not greater than 15.

Yes, the majority of the conditions are met.

Answer 1

To determine if the sides with lengths of 5 inches, 10 inches, and 15 inches can form a triangle, we must apply the Triangle Inequality Theorem. This theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following conditions must be met:

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

For the sides given:

• \( a = 5 \)
• \( b = 10 \)
• \( c = 15 \)

Let's check each condition:

1. \( 5 + 10 > 15 \) → \( 15 > 15 \) (not true)
2. \( 5 + 15 > 10 \) → \( 20 > 10 \) (true)
3. \( 10 + 15 > 5 \) → \( 25 > 5 \) (true)

Since the first condition fails (the sum of the lengths of the two shorter sides is not greater than the length of the longest side), these sides cannot form a triangle.

Therefore, the correct response is:

No, because the sum of 5 + 10 is not greater than 15.