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 15 inches 15 inches 20 inches 20 inches 3 inches

Answer 1

To determine the possible lengths of the third side of a triangle when the lengths of the other two sides are known, we can use the Triangle Inequality Theorem. This theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \):

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

Let's denote the lengths of the sides as follows:

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

Now we can apply the theorem to find the valid range for \( c \):

1. From \( a + b > c \): \[ 14 + 6 > c \implies 20 > c \implies c < 20 \]

2. From \( a + c > b \): \[ 14 + c > 6 \implies c > 6 - 14 \implies c > -8 \text{ (which is always true)} \]

3. From \( b + c > a \): \[ 6 + c > 14 \implies c > 14 - 6 \implies c > 8 \]

So the two inequalities we need to satisfy are:

• \( c > 8 \)
• \( c < 20 \)

Thus, the range for \( c \) is: \[ 8 < c < 20 \]

Now let's evaluate the possible responses:

• 7 inches: Not valid (less than 8).
• 15 inches: Valid (within 8 and 20).
• 20 inches: Not valid (equal to 20).
• 3 inches: Not valid (less than 8).

Therefore, the only valid option for the length of the third side is 15 inches.