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

11
11

2
2

1
1

9

Answer 1

To determine which option could be the third side of a triangle when the first two sides measure 4 and 6, we can use the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let the lengths of the sides be represented as follows:
- Side 1 = 4
- Side 2 = 6
- Side 3 = x (the unknown side)

According to the Triangle Inequality Theorem, we have the following conditions:

1. \( 4 + 6 > x \) (or \( 10 > x \) ⇒ \( x < 10 \))
2. \( 4 + x > 6 \) (or \( x > 2 \))
3. \( 6 + x > 4 \) (This condition is always satisfied if \( x > 2 \))

From the conditions, we can deduce that:
- \( x \) must be greater than 2 and less than 10.

Now, let's analyze the given options:

1. **11**: This does not satisfy \( x < 10 \).
2. **2**: This does not satisfy \( x > 2 \).
3. **1**: This does not satisfy \( x > 2 \).
4. **9**: This satisfies both conditions \( 2 < 9 < 10 \).

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