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)

Question

Responses

3 inches
3 inches

15 inches
15 inches

7 inches
7 inches

20 inches

Answer 1

To determine which lengths could be the length of the third side of the triangle, we can use the triangle inequality theorem. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Let the lengths of the sides be:

• \( a = 14 \) inches
• \( b = 6 \) inches
• \( c \) = length of the third side

According to the triangle inequality theorem, we have the following conditions:

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

Substituting the values we have:

1. \( 14 + 6 > c \) → \( 20 > c \) → \( c < 20 \)
2. \( 14 + c > 6 \) → \( c > 6 - 14 = -8 \), which is always true since c must be positive.
3. \( 6 + c > 14 \) → \( c > 14 - 6 = 8 \)

Combining these results, we find:

• From the first inequality, \( c < 20 \)
• From the third inequality, \( c > 8 \)

Therefore, the possible range for the third side \( c \) is:

\[ 8 < c < 20 \]

Now we can evaluate the given options:

1. 3 inches: Not valid (3 < 8)
2. 15 inches: Valid (8 < 15 < 20)
3. 7 inches: Not valid (7 < 8)
4. 20 inches: Not valid (20 is not less than 20)

The only valid length for the third side is 15 inches.