If x is a positive integer, and two sides of a certain triangle have lengths x+2 and x+2 respectively, which of the following could be the length of the third side of the triangle?
I. x + 1
II. x + 5
III. 2x + 1
Answers.
A) I only
B) II only
C) I and II only
D) I and III only
E) I, II and III
Let the third side of triangle be AB.   Triangle rule states that:   Difference of other 2 sides &lt; any side &lt; sum of other two sides   So x+2 - (x+2) &lt; AB &lt; x+2 + (x+2)   i.e. 0 &lt; AB &lt; 2x + 4   Let's take x =1 :   So 0 &lt; AB &lt; 6   Check if any of the answers are between 0 and 4.   I: x+1 = 2 ; could be a possible value.   II: x+5 = 6 ; could not be a possible value.   III: 2x+1 = 3; could be a possible value.   I, II and III could not be a possible value.   Answer: I and III only  2 +1= 
<gadget id="calculator">  2 +1</gadget>
<output>3</output>
3; could be a possible value.   I, II and III could not be a possible value.   Answer: I and III only
<result>D</result>
If x is a positive integer, and two sides of a certain triangle have lengths x+2 and x+2 respectively, which of the following could be the length of the third side of the triangle?
I. x + 1
II. x + 5
III. 2x + 1