anne has four sticks of different lengths. the table shows the lengths of the sticks. stick a b c d length 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 be more than one answer. (2 points) responses triangle abc triangle abc triangle abd triangle abd triangle bcd triangle bcd triangle acd

Answer 1

To determine which combinations of sticks Anne can use to make a triangle, we can use the triangle inequality theorem. This theorem states that for any three lengths \( a \), \( b \), and \( c \) to form a triangle, the following must be true:

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

Given the stick lengths:

• Stick A = 3 inches
• Stick B = 5 inches
• Stick C = 6 inches
• Stick D = 10 inches

Let's analyze each combination of three sticks:

1. Combination ABC (3, 5, 6):

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

2. Combination ABD (3, 5, 10):

   • \( 3 + 5 = 8 \not> 10 \) (False) This combination cannot form a triangle.

3. Combination ACD (3, 6, 10):

   • \( 3 + 6 = 9 \not> 10 \) (False) This combination cannot form a triangle.

4. Combination BCD (5, 6, 10):

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

Based on the above evaluations, the possible combinations of sticks that can form a triangle are:

• Triangle ABC
• Triangle BCD

Thus, the valid responses are:

• Triangle ABC
• Triangle BCD