Question

Richard draws a triangle in the dirt. Two of the sides of the triangle measure 14 inches and 6 inches. Which could be the length of the third side of Richard’s triangle?(1 point)
Responses

7 inches
7 inches

20 inches
20 inches

15 inches
15 inches

3 inches

Answer 1

To determine which lengths could be the length of the third side of a triangle given the other two sides, 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’s denote the sides of the triangle as follows:
- \( a = 14 \) inches (first side)
- \( b = 6 \) inches (second side)
- \( c \) (third side, which we need to find)

According to the triangle inequality theorem, the following must be true:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Applying these inequalities:

1. \( 14 + 6 > c \) → \( 20 > c \) → \( c < 20 \)
2. \( 14 + c > 6 \) → \( c > -8 \) (this inequality does not impose a restriction since \( c \) must be positive)
3. \( 6 + c > 14 \) → \( c > 8 \)

Now we combine the conditions:
- From the first inequality, we find \( c < 20 \).
- From the third inequality, we find \( c > 8 \).

So, the length of the third side \( c \) must satisfy:

\[ 8 < c < 20 \]

Now examining the given options:
- **7 inches**: Does not satisfy \( c > 8 \) (not possible)
- **20 inches**: Does not satisfy \( c < 20 \) (not possible)
- **15 inches**: Satisfies \( 8 < 15 < 20 \) (possible)
- **3 inches**: Does not satisfy \( c > 8 \) (not possible)

The only option that could be the length of the third side is **15 inches**.