Question

Apply the Triangle Inequality Theorem and determine which option could be the third side of a triangle if the first two sides measure 4 and 6.(1 point)
Responses

9
9

2
2

1
1

11
11

Answer 1

The Triangle Inequality Theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following inequalities must hold:

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

Given sides of lengths 4 and 6, let's denote the unknown third side as \( x \). We need to apply the inequalities:

1. \( 4 + 6 > x \) → \( 10 > x \) → \( x < 10 \)
2. \( 4 + x > 6 \) → \( x > 2 \)
3. \( 6 + x > 4 \) → \( x > -2 \) (This is always true if \( x > 2 \))

From these inequalities, we have:
- \( x < 10 \)
- \( x > 2 \)

Therefore, the value of \( x \) must be in the range \( 2 < x < 10 \).

Now, let's examine the options:

- **9**: This is within the range \( 2 < x < 10 \). (Valid)
- **2**: This is not greater than 2. (Not valid)
- **1**: This is not within the range. (Not valid)
- **11**: This is greater than 10. (Not valid)

The only option that could be the third side of the triangle is **9**.