Examkin

Question

Difficulty: EasyTriangles: Properties, Perimeter, and Area

Two sides of a triangle have lengths of 55 centimeters and 99 centimeters. Which of the following could be the length, in centimeters, of the third side? Select all such lengths.

  1. A
    44
  2. 66Answer
  3. 1010Answer
  4. D
    1414
  5. E
    1616

Answer

The possible lengths for the third side are 66 centimeters and 1010 centimeters.
According to the Triangle Inequality Theorem, the length of the third side xx must be strictly greater than the difference of the two given sides (9−5=49 - 5 = 4) and strictly less than their sum (9+5=149 + 5 = 14). This establishes the valid range as 4<x<144 < x < 14. The options 66 centimeters and 1010 centimeters are the only values provided that fall strictly within this interval.

Step-by-Step Solution

1
Apply the Triangle Inequality Theorem to determine the allowable range for the third side.
The length of the third side, xx, must satisfy ∣9−5∣<x<9+5|9 - 5| < x < 9 + 5, which simplifies to 4<x<144 < x < 14.
The sum of the lengths of any two sides of a non-degenerate triangle must be strictly greater than the length of the remaining side.
2
Check each candidate option against the inequality 4<x<144 < x < 14.
The values 66 and 1010 satisfy 4<x<144 < x < 14, whereas 44, 1414, and 1616 do not.
Only numbers strictly within the open interval (4,14)(4, 14) can form a valid triangle with side lengths 55 and 99.

Key Concept

Triangle Inequality Theorem
Rate this question

Topics

Triangles: Properties, Perimeter, and Area

More questions from this topic