Question

Joanna has four straws of different lengths. The table shows the lengths of the straws. Straw A B C D Length in inches 7 9 3 10Joanna makes as many triangles as she can using combinations of three of these straws. Determine how many different triangles she can make. What combinations of straws does she use? Show your work.(4 points)

Answer 1

To determine how many different triangles Joanna can make using the straws with lengths A (7 inches), B (9 inches), C (3 inches), and D (10 inches), we need to use the triangle inequality theorem. This theorem states that for three sides to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

First, let's list all the combinations of three straws that Joanna can choose from the four straws:

1. A, B, C (7, 9, 3)
2. A, B, D (7, 9, 10)
3. A, C, D (7, 3, 10)
4. B, C, D (9, 3, 10)

Now, we will evaluate each combination using the triangle inequality theorem:

1. Combination A, B, C (7, 9, 3):

   • \(7 + 3 = 10\) (not greater than 9)
   • \(7 + 9 = 16 > 3\)
   • \(9 + 3 = 12 > 7\)
   • Does not form a triangle.

2. Combination A, B, D (7, 9, 10):

   • \(7 + 9 = 16 > 10\)
   • \(7 + 10 = 17 > 9\)
   • \(9 + 10 = 19 > 7\)
   • Forms a triangle.

3. Combination A, C, D (7, 3, 10):

   • \(7 + 3 = 10\) (not greater than 10)
   • \(7 + 10 = 17 > 3\)
   • \(3 + 10 = 13 > 7\)
   • Does not form a triangle.

4. Combination B, C, D (9, 3, 10):

   • \(9 + 3 = 12 > 10\)
   • \(9 + 10 = 19 > 3\)
   • \(3 + 10 = 13 > 9\)
   • Forms a triangle.

From our analysis, Joanna can form triangles from the following combinations:

1. A, B, D (7, 9, 10)
2. B, C, D (9, 3, 10)

In summary, Joanna can create 2 different triangles using the combinations:

• Combination 1: Straws A, B, D (7, 9, 10)
• Combination 2: Straws B, C, D (9, 3, 10)

Therefore, the answer is 2 different triangles and the combinations are (7, 9, 10) and (9, 3, 10).

Answer 2

What about 7, 3, 9?