To determine how many different triangles can be formed using three of Joanna's straws, we must use the triangle inequality theorem, which states that for any three lengths \( a \), \( b \), and \( c \) (where \( a \leq b \leq c \)), the following conditions must hold:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
In the case of straws, since all straws have positive lengths, the first condition \( a + b > c \) is sufficient to verify that three lengths can form a triangle because the other two conditions will be satisfied if the first is.
The lengths of the straws are:
- \( A = 7 \) inches
- \( B = 9 \) inches
- \( C = 3 \) inches
- \( D = 10 \) inches
Now, let's list all combinations of three straws and check if they can form a triangle.
-
Combination: 7, 9, 3
- Check: \( 3 + 7 > 9 \) (False)
- Cannot form a triangle.
-
Combination: 7, 9, 10
- Check: \( 7 + 9 > 10 \) (True)
- Can form a triangle.
-
Combination: 7, 3, 10
- Check: \( 3 + 7 > 10 \) (False)
- Cannot form a triangle.
-
Combination: 9, 3, 10
- Check: \( 3 + 9 > 10 \) (False)
- Cannot form a triangle.
From our checks, we see that the only combination that can form a triangle is the combination of the 7, 9, and 10 inch straws.
So, Joanna can make 1 different triangle using the following combination of straws:
- 7 inches, 9 inches, 10 inches.
1 answer