Alfred has three pieces of wood. Board A is 11 inches long, board B is 4 inches long, and board C is 6 inches long. If the full length of each board is used, can the three pieces of wood be placed together to form a triangle?

Question

A.
yes, 4 + 6 < 11
B.
no, 11 - 4 < 6
C.
no, 4 + 6 < 11
D.
yes, 11 - 6 > 4

Answer 1

To determine if three lengths can form a triangle, we can use the triangle inequality theorem. This theorem states that for any three sides \( a \), \( b \), and \( c \) of a triangle, the following conditions must be true:

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

In this case, the lengths of the boards are:

• Board A: 11 inches (let's denote it as \( a \))
• Board B: 4 inches (let's denote it as \( b \))
• Board C: 6 inches (let's denote it as \( c \))

Now we will check the three inequalities:

1. \( a + b > c \): \( 11 + 4 > 6 \) \( 15 > 6 \) (True)

2. \( a + c > b \): \( 11 + 6 > 4 \) \( 17 > 4 \) (True)

3. \( b + c > a \): \( 4 + 6 > 11 \) \( 10 > 11 \) (False)

Since the third condition \( b + c > a \) is false (10 is not greater than 11), the three pieces of wood cannot be arranged to form a triangle.

Thus, the correct answer is:

C. no, 4 + 6 < 11