Question

which of the following sets of numbers could be the lengths of the sides of a triangle? select three correct answers. 16, 47, 19 16, 20, 18 19, 14, 17 4, 16, 13 20, 39, 17 23, 20, 2

Explanation:

Step1: Recall triangle - inequality theorem

For three side - lengths \(a\), \(b\), and \(c\) of a triangle, the following three inequalities must hold: \(a + b>c\), \(a + c>b\), and \(b + c>a\).

Step2: Check the set \((16,47,19)\)

\(16+19 = 35<47\), so it cannot form a triangle.

Step3: Check the set \((16,20,18)\)

\(16 + 20=36>18\), \(16+18 = 34>20\), \(20 + 18=38>16\). It can form a triangle.

Step4: Check the set \((19,14,17)\)

\(19+14 = 33>17\), \(19 + 17=36>14\), \(14 + 17=31>19\). It can form a triangle.

Step5: Check the set \((4,16,13)\)

\(4+13 = 17>16\), \(4+16 = 20>13\), \(13 + 16=29>4\). It can form a triangle.

Step6: Check the set \((20,39,17)\)

\(20+17 = 37<39\), so it cannot form a triangle.

Step7: Check the set \((23,20,2)\)

\(20 + 2=22<23\), so it cannot form a triangle.

Answer:

16, 20, 18; 19, 14, 17; 4, 16, 13