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Difficulty: MediumTriangle Properties and Angle Theorems

A triangle has two sides of length 5 and 12. The third side has a length of xx, where xx is an integer. If the perimeter of the triangle is a multiple of 5, what is the sum of all possible values of xx?

  1. A
    13
  2. B
    25
  3. 21Answer
  4. D
    24
  5. E
    39

Answer

The sum of all possible values of the third side length is 21.
To find the sum of all possible values of xx, we first apply the Triangle Inequality Theorem. For a triangle with side lengths 5, 12, and xx, the third side must satisfy 12−5<x<12+512 - 5 < x < 12 + 5, which simplifies to 7<x<177 < x < 17. The perimeter PP of the triangle is given by P=5+12+x=17+xP = 5 + 12 + x = 17 + x. Given that 7<x<177 < x < 17, the perimeter must be between 17+7=2417 + 7 = 24 and 17+17=3417 + 17 = 34. The only multiples of 5 within this range are 25 and 30. Setting the perimeter equal to these values gives 17+x=25  ⟹  x=817 + x = 25 \implies x = 8, and 17+x=30  ⟹  x=1317 + x = 30 \implies x = 13. Both values are integers and satisfy the triangle inequality. The sum of these values is 8+13=218 + 13 = 21.

Step-by-Step Solution

1
Apply the Triangle Inequality Theorem to find the range of possible values for the third side, xx.
12−5<x<12+512 - 5 < x < 12 + 5, which simplifies to 7<x<177 < x < 17.
The length of any side of a triangle must be strictly greater than the difference between the other two sides and strictly less than their sum.
2
Determine the expression for the perimeter of the triangle and find the bounds for the perimeter.
Perimeter P=5+12+x=17+xP = 5 + 12 + x = 17 + x. Since 7<x<177 < x < 17, the perimeter must satisfy 17+7<P<17+1717 + 7 < P < 17 + 17, which means 24<P<3424 < P < 34.
The perimeter of a triangle is the sum of its three side lengths.
3
Identify which values of the perimeter in this range are multiples of 5, and find the corresponding values of xx.
The multiples of 5 between 24 and 34 are 25 and 30. If P=25P = 25, then 17+x=25  ⟹  x=817 + x = 25 \implies x = 8. If P=30P = 30, then 17+x=30  ⟹  x=1317 + x = 30 \implies x = 13. Both x=8x = 8 and x=13x = 13 are integers that satisfy the initial inequality.
We must find the integer values of xx that make the perimeter a multiple of 5.
4
Calculate the sum of all possible values of xx.
8+13=218 + 13 = 21.
The question asks for the sum of all valid integer values of xx.

Key Concept

Triangle Inequality Theorem and Perimeter Calculations
Estimated Time:1m 30s
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Topics

Triangle Properties and Angle Theorems

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