Anne has 4 sticks of different lengths. The table shows the lengths of the sticks.

Question

Stick A B C D
Lengths in Inches 3 5 6 10

Anne makes as many triangles as she can using combinations of three sticks.

Choose ALL possible combinations of sticks she can use to make a triangle. There may ne more than one answer.

a. Triangle BCD
b. Triangle ABC
c. Triangle ACD
d. Triangle ABD

Answer 1

To determine if a combination of sticks can form a triangle, we can use the triangle inequality theorem. According to this theorem, for any three sides a, b, and c, the following conditions must be satisfied:

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

Let's examine each combination of sticks proposed to see if they can form a triangle:

1. Triangle BCD (Lengths: 5, 6, 10)

   • Check: \( 5 + 6 > 10 \) (11 > 10) True
   • Check: \( 5 + 10 > 6 \) (15 > 6) True
   • Check: \( 6 + 10 > 5 \) (16 > 5) True
   • This combination can form a triangle.

2. Triangle ABC (Lengths: 3, 5, 6)

   • Check: \( 3 + 5 > 6 \) (8 > 6) True
   • Check: \( 3 + 6 > 5 \) (9 > 5) True
   • Check: \( 5 + 6 > 3 \) (11 > 3) True
   • This combination can form a triangle.

3. Triangle ACD (Lengths: 3, 6, 10)

   • Check: \( 3 + 6 > 10 \) (9 > 10) False
   • Check: \( 3 + 10 > 6 \) (13 > 6) True
   • Check: \( 6 + 10 > 3 \) (16 > 3) True
   • This combination cannot form a triangle.

4. Triangle ABD (Lengths: 3, 5, 10)

   • Check: \( 3 + 5 > 10 \) (8 > 10) False
   • Check: \( 3 + 10 > 5 \) (13 > 5) True
   • Check: \( 5 + 10 > 3 \) (15 > 3) True
   • This combination cannot form a triangle.

In summary, the possible combinations of sticks that can form a triangle are:

• a. Triangle BCD
• b. Triangle ABC

Therefore, the correct answers are a and b.