Question

If a triangle has sides that measure 6 cm and 10 cm, which of the following could be the third side?
Option #1:3 cm
Option #2: 9 cm
Option #3: 16 cm
Option #4: 20 cm

Answer 1

To determine the possible lengths of the third side of a triangle when two sides are known, 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.

Given sides of lengths \( a = 6 \) cm and \( b = 10 \) cm, we can denote the length of the third side as \( c \). The triangle inequalities can be expressed as:

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

Substituting the known values into these inequalities:

1. \( 6 + 10 > c \)
- \( 16 > c \)
- \( c < 16 \)

2. \( 6 + c > 10 \)
- \( c > 4 \)

3. \( 10 + c > 6 \)
- This inequality is always true for positive \( c \).

Putting together these results, we find the range for \( c \) is:

\[
4 < c < 16
\]

Now, we can evaluate the options provided:

- Option #1: 3 cm (not valid since \( c > 4 \))
- Option #2: 9 cm (valid since \( 4 < 9 < 16 \))
- Option #3: 16 cm (not valid since \( c < 16 \))
- Option #4: 20 cm (not valid since \( c < 16 \))

Therefore, the only valid option for the length of the third side is:

**Option #2: 9 cm**.