Full solution

Q. Which of the following sets of numbers could represent the three sides of a triangle?

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\{12,19,31\}

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\{10,25,37\}

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\{10,15,26\}

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\{7,21,27\}

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.
Apply Theorem to First Set: Apply the Triangle Inequality Theorem to the first set ${12, 19, 31}$ . Check if the sum of the two smallest numbers is greater than the largest number: $12 + 19 > 31$ ?
Calculate First Set: Perform the calculation for the first set: $12 + 19 = 31$ . Since $31$ is not greater than $31$ , the first set does not satisfy the Triangle Inequality Theorem.
Apply Theorem to Second Set: Apply the Triangle Inequality Theorem to the second set $\{10, 25, 37\}$ . Check if the sum of the two smallest numbers is greater than the largest number: $10 + 25 > 37$ ?
Calculate Second Set: Perform the calculation for the second set: $10 + 25 = 35$ . Since $35$ is not greater than $37$ , the second set does not satisfy the Triangle Inequality Theorem.
Apply Theorem to Third Set: Apply the Triangle Inequality Theorem to the third set $\{10, 15, 26\}$ . Check if the sum of the two smallest numbers is greater than the largest number: $10 + 15 > 26$ ?
Calculate Third Set: Perform the calculation for the third set: $10 + 15 = 25$ . Since $25$ is not greater than $26$ , the third set does not satisfy the Triangle Inequality Theorem.
Apply Theorem to Fourth Set: Apply the Triangle Inequality Theorem to the fourth set $\{7, 21, 27\}$ . Check if the sum of the two smallest numbers is greater than the largest number: $7 + 21 > 27$ ?
Calculate Fourth Set: Perform the calculation for the fourth set: $7 + 21 = 28$ . Since $28$ is greater than $27$ , the fourth set satisfies the Triangle Inequality Theorem.
Conclude Valid Triangle Set: Conclude that the fourth set $\{7, 21, 27\}$ is the only set that could represent the sides of a triangle.