The measures of the lengths of the three sides of a triangle are prime numbers. If two of the sides are 5 and 23, what is one possible value for the length of the third side?
Correct Answer: 19 or 23
Since two of the sides of the triangle are 5 and 23, let y represent the third side. By the triangle inequality
- 5 + y > 23 or y > 18
- 5 + 23 > y or y < 28
- 23 + y > 5 or y > -18
The lengths of the three sides of the triangle are prime numbers so that y must be a prime number between 18 and 28. Thus, the two possible values for the length of the third side of the triangle are 19 and 23. Either answer may be gridded as the correct answer to the problem.
(from the October 17, 1998 test)