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Which of the following sets of numbers could represent the three sides of a triangle? {4,10,12} {4,17,21} {4,14,20} {13,18,31}

Which of the following sets of numbers could represent the three sides of a triangle?\newline{4,10,12} \{4,10,12\} \newline{4,17,21} \{4,17,21\} \newline{4,14,20} \{4,14,20\} \newline{13,18,31} \{13,18,31\}

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Q. Which of the following sets of numbers could represent the three sides of a triangle?\newline{4,10,12} \{4,10,12\} \newline{4,17,21} \{4,17,21\} \newline{4,14,20} \{4,14,20\} \newline{13,18,31} \{13,18,31\}
  1. Recall Triangle Inequality Theorem: Recall the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
  2. Apply Theorem to Set {4,10,12}\{4,10,12\}: Apply the Triangle Inequality Theorem to the first set {4,10,12}\{4,10,12\}. Check if 4+10>124 + 10 > 12, 4+12>104 + 12 > 10, and 10+12>410 + 12 > 4. Calculations: 4+10=144 + 10 = 14, 4+12=164 + 12 = 16, 10+12=2210 + 12 = 22. Since 14>1214 > 12, 16>1016 > 10, and {4,10,12}\{4,10,12\}00, all conditions are satisfied.
  3. Apply Theorem to Set {4,17,21}\{4,17,21\}: Apply the Triangle Inequality Theorem to the second set {4,17,21}\{4,17,21\}. Check if 4+17>214 + 17 > 21, 4+21>174 + 21 > 17, and 17+21>417 + 21 > 4. Calculations: 4+17=214 + 17 = 21, 4+21=254 + 21 = 25, 17+21=3817 + 21 = 38. Since 2121 is not greater than 2121, the condition 4+17>214 + 17 > 21 is not satisfied.
  4. Apply Theorem to Set {4,14,20}\{4,14,20\}: Apply the Triangle Inequality Theorem to the third set {4,14,20}\{4,14,20\}. Check if 4+14>204 + 14 > 20, 4+20>144 + 20 > 14, and 14+20>414 + 20 > 4. Calculations: 4+14=184 + 14 = 18, 4+20=244 + 20 = 24, 14+20=3414 + 20 = 34. Since 1818 is not greater than 2020, the condition 4+14>204 + 14 > 20 is not satisfied.
  5. Apply Theorem to Set {13,18,31}\{13,18,31\}: Apply the Triangle Inequality Theorem to the fourth set {13,18,31}\{13,18,31\}. Check if 13+18>3113 + 18 > 31, 13+31>1813 + 31 > 18, and 18+31>1318 + 31 > 13. Calculations: 13+18=3113 + 18 = 31, 13+31=4413 + 31 = 44, 18+31=4918 + 31 = 49. Since 3131 is not greater than 3131, the condition 13+18>3113 + 18 > 31 is not satisfied.

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