Two sides of a triangle measure 4 and 12. Which of the following could equal the length of the third side?

Options

A. 5
B. 7
C. 9
D. 20

Answer

To determine which of the given options could be the length of the third side of a triangle with two sides measuring 4 and 12, we will 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.

### Step-by-Step Solution:

1. **Identify the sides of the triangle:**
We have two sides of the triangle: \( a = 4 \) and \( b = 12 \). We need to find the possible length of the third side \( c \).

2. **Apply the triangle inequality theorem:**
According to the triangle inequality theorem, we have three conditions to check:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)

3. **Check each option one by one:**

**Option 1: \( c = 5 \)**
- Check \( 4 + 5 > 12 \):
\( 9 > 12 \) (False)
- Since this condition is false, **5 cannot be the length of the third side**.

**Option 2: \( c = 7 \)**
- Check \( 4 + 7 > 12 \):
\( 11 > 12 \) (False)
- Since this condition is false, **7 cannot be the length of the third side**.

**Option 3: \( c = 9 \)**
- Check \( 4 + 9 > 12 \):
\( 13 > 12 \) (True)
- Check \( 4 + 12 > 9 \):
\( 16 > 9 \) (True)
- Check \( 12 + 9 > 4 \):
\( 21 > 4 \) (True)
- All conditions are satisfied, so **9 can be the length of the third side**.

**Option 4: \( c = 20 \)**
- Check \( 4 + 20 > 12 \):
\( 24 > 12 \) (True)
- Check \( 4 + 12 > 20 \):
\( 16 > 20 \) (False)
- Since this condition is false, **20 cannot be the length of the third side**.

4. **Conclusion:**
After checking all options, only **option 3 (9)** satisfies the triangle inequality conditions. Therefore, the length of the third side can be 9.