Question

joanna has four straws of different lengths. the table shows the lengths of the straws. length of straws straw a b c d length in inches 7 9 3 10 joanna 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)

Explanation:

Step1: Recall triangle - inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Step2: List all combinations of three - straw lengths

We have 4 straws, and the number of combinations of choosing 3 out of 4 is given by the combination formula $C(n,k)=\frac{n!}{k!(n - k)!}$, where $n = 4$ and $k=3$. Here, $C(4,3)=\frac{4!}{3!(4 - 3)!}=\frac{4!}{3!1!}=4$ combinations. The combinations are: $(A,B,C)$ with lengths $(7,9,3)$; $(A,B,D)$ with lengths $(7,9,10)$; $(A,C,D)$ with lengths $(7,3,10)$; $(B,C,D)$ with lengths $(9,3,10)$.

Step3: Check each combination using the triangle - inequality theorem

• For $(7,9,3)$:
• $7 + 9>3$ (True), $7+3 > 9$ (True), $9 + 3>7$ (True). So, a triangle can be formed.
• For $(7,9,10)$:
• $7+9>10$ (True), $7 + 10>9$ (True), $9+10>7$ (True). So, a triangle can be formed.
• For $(7,3,10)$:
• $7+3=10$ (False). So, a triangle cannot be formed.
• For $(9,3,10)$:
• $9+3>10$ (True), $9 + 10>3$ (True), $3+10>9$ (True). So, a triangle can be formed.

Answer:

She can make 3 different triangles. The combinations are: Straws A, B, C (lengths 7, 9, 3 inches); Straws A, B, D (lengths 7, 9, 10 inches); Straws B, C, D (lengths 9, 3, 10 inches).