Question

select the correct answer. based on these segment lengths, which group of segments can form a triangle? a. 3, 10, 14 b. 8, 7, 13 c. 3, 2, 5 d. 20, 7, 13

Explanation:

Step1: Recall Triangle Inequality Theorem

For three side lengths \(a\), \(b\), \(c\) (where \(c\) is the largest), the theorem states \(a + b>c\), \(a + c>b\), and \(b + c>a\). We can simplify to check the sum of the two smaller sides > largest side.

Step2: Check Option A (3, 10, 14)

Smaller sides: 3, 10. Sum: \(3 + 10 = 13\). Largest side: 14. \(13<14\) → Not a triangle.

Step3: Check Option B (8, 7, 13)

Smaller sides: 7, 8. Sum: \(7 + 8 = 15\). Largest side: 13. \(15>13\) → Check other inequalities (7 + 13 > 8, 8 + 13 > 7, which hold). So possible triangle.

Step4: Check Option C (3, 2, 5)

Smaller sides: 2, 3. Sum: \(2 + 3 = 5\). Largest side: 5. \(5 = 5\) (needs to be >) → Not a triangle.

Step5: Check Option D (20, 7, 13)

Smaller sides: 7, 13. Sum: \(7 + 13 = 20\). Largest side: 20. \(20 = 20\) (needs to be >) → Not a triangle.

Answer:

B. 8, 7, 13