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)

Question

Responses

1
1

2
2

11
11

9

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. \(b + c > a\)

In this case, we are given two sides \(a = 4\) and \(b = 6\), and we need to determine valid values for the third side \(c\).

1. From \(a + b > c\): \[ 4 + 6 > c \implies 10 > c \implies c < 10 \]

2. From \(a + c > b\): \[ 4 + c > 6 \implies c > 2 \]

3. From \(b + c > a\): \[ 6 + c > 4 \implies c > -2 \quad (\text{This is always true for positive } c) \]

Combining the inequalities from the first two steps, we get: \[ 2 < c < 10 \]

Now, we can evaluate the options provided:

• Option 1: 1

  • Not valid, since \(1 < 2\).

• Option 2: 2

  • Not valid, since \(2\) is not greater than \(2\).

• Option 3: 11

  • Not valid, since \(11 > 10\).

• Option 4: 9

  • Valid, since \(2 < 9 < 10\).

Therefore, the only option that could be the third side of the triangle is:

9