Question

A triangle has side lengths of 12 inches and 3 inches.
Sort the following lengths in inches by which could be the length of the third side of this triangle.
Could be the third side length Could not be the third side length
17
16.5
15
14
19
13

Answer 1

To determine which lengths could be the third side of a triangle given the other two sides (12 inches and 3 inches), we can use the Triangle Inequality Theorem. According to this theorem, for any triangle with sides \( a \), \( b \), and \( c \):

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

In this scenario, we can denote the given sides as \( a = 12 \) inches and \( b = 3 \) inches, and we need to find the range for the third side \( c \).

Applying the Triangle Inequality:

1. \( 12 + 3 > c \)
- This simplifies to \( c < 15 \).

2. \( 12 + c > 3 \)
- This simplifies to \( c > -9 \) (which is always true since side lengths are positive).

3. \( 3 + c > 12 \)
- This simplifies to \( c > 9 \).

From these inequalities, we find that:
- The length of the third side \( c \) must be greater than 9 and less than 15.

Thus, the valid range for the third side length is:
\( 9 < c < 15 \).

Now we can sort the given lengths based on this range.

**Could be the third side length:**
- 13
- 14

**Could not be the third side length:**
- 17
- 16.5
- 15
- 19

So the sorted lists are:

**Could be the third side length:** 13, 14
**Could not be the third side length:** 17, 16.5, 15, 19