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Difficulty: MediumTriangle Properties and Angle Theorems

A triangular framework is being constructed using three metal rods. Two of the rods have lengths of 1414 inches and 2525 inches. The third rod must have an integer length of xx inches. What is the number of possible integer values for xx?

  1. A
    14
  2. B
    28
  3. 27Answer
  4. D
    29
  5. E
    39

Answer

27
The correct answer is the value representing 27 possible integers. By the Triangle Inequality Theorem, the third side xx must satisfy the inequality 25−14<x<25+1425 - 14 < x < 25 + 14, which simplifies to 11<x<3911 < x < 39. Since xx must be an integer, it can take any value from 1212 to 3838, inclusive. Counting these integers yields 38−12+1=2738 - 12 + 1 = 27 possible values.

Step-by-Step Solution

1
Apply the Triangle Inequality Theorem to set up the inequality for the third side xx.
25−14<x<25+1425 - 14 < x < 25 + 14
The Triangle Inequality Theorem states that the length of any side of a triangle must be strictly greater than the difference of the other two sides and strictly less than their sum.
2
Simplify the compound inequality to find the bounds for xx.
11<x<3911 < x < 39
Subtracting and adding the side lengths gives the range of possible values for the third side.
3
Identify the set of integers that satisfy the simplified inequality.
x∈{12,13,14,…,37,38}x \in \{12, 13, 14, \dots, 37, 38\}
Since xx must be an integer and the inequalities are strict, the minimum integer value is 1212 and the maximum is 3838.
4
Calculate the count of integers in the range from 1212 to 3838, inclusive.
38−12+1=2738 - 12 + 1 = 27
To find the number of integers in an inclusive range [a,b][a, b], use the formula b−a+1b - a + 1.

Key Concept

Triangle Inequality Theorem

Alternative Method

To find the number of integers strictly between two integers aa and bb (where a<ba < b), you can use the formula (b−a)−1(b - a) - 1. For the bounds 11<x<3911 < x < 39, the calculation is (39−11)−1=28−1=27(39 - 11) - 1 = 28 - 1 = 27.
Estimated Time:1m 15s
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Topics

Triangle Properties and Angle Theorems

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