Question

ab = 5, bc = 7, cd = 10, de = 8, and ef = 12
which of the following could form a triangle? select all that apply
a ab, bc and cd
b ab, bc and de
c ab, bc and ef
d bc, cd and ef
e ab, de and ef

Explanation:

To determine if three lengths can form a triangle, we use the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the remaining side.

Step 1: Analyze Option A (AB = 5, BC = 7, CD = 10)

Check the three inequalities:

• \(5 + 7 > 10\): \(12 > 10\) (True)
• \(5 + 10 > 7\): \(15 > 7\) (True)
• \(7 + 10 > 5\): \(17 > 5\) (True)

All inequalities hold, so AB, BC, and CD can form a triangle.

Step 2: Analyze Option B (AB = 5, BC = 7, DE = 8)

Check the three inequalities:

• \(5 + 7 > 8\): \(12 > 8\) (True)
• \(5 + 8 > 7\): \(13 > 7\) (True)
• \(7 + 8 > 5\): \(15 > 5\) (True)

All inequalities hold, so AB, BC, and DE can form a triangle.

Step 3: Analyze Option C (AB = 5, BC = 7, EF = 12)

Check the inequality \(5 + 7 > 12\): \(12 > 12\) (False, since it must be greater, not equal). So AB, BC, and EF cannot form a triangle.

Step 4: Analyze Option D (BC = 7, CD = 10, EF = 12)

Check the three inequalities:

• \(7 + 10 > 12\): \(17 > 12\) (True)
• \(7 + 12 > 10\): \(19 > 10\) (True)
• \(10 + 12 > 7\): \(22 > 7\) (True)

All inequalities hold, so BC, CD, and EF can form a triangle.

Step 5: Analyze Option E (AB = 5, DE = 8, EF = 12)

Check the inequality \(5 + 8 > 12\): \(13 > 12\) (True)

• \(5 + 12 > 8\): \(17 > 8\) (True)
• \(8 + 12 > 5\): \(20 > 5\) (True)

All inequalities hold, so AB, DE, and EF can form a triangle.

Answer:

A. AB, BC and CD
B. AB, BC and DE
D. BC, CD and EF
E. AB, DE and EF