Question

In a triangle JKL, the length of two sides is given as 8 and 13 and the length of third side is unknown. Which of the following could be the length of this unknown side? Indicate all possible options.

• A. 2
• B. 4
• C. 6
• D. 9
• E. 15
• F. 22

Hint:

Always apply the triangle inequality: for any triangle with sides \(a, b, c\), each side must satisfy \(|a-b|<c<a+b\).

Answer 1

The Correct Option is B

Step 1: Recall the triangle inequality theorem.
In any triangle, the length of one side must be less than the sum of the other two sides and greater than their difference.
So if the given two sides are 8 and 13, then for the unknown side \( x \): \[ |13 - 8|<x<13 + 8 \]

Step 2: Simplify the inequality.
\[ 5<x<21 \]

Step 3: Check the given options.
- (A) \(2\): Not valid, since \(2<5\).
- (B) \(4\): Not valid, since \(4<5\). Wait correction → \(4\) is also less than 5, so not valid.
- (C) \(6\): Valid, since \(6\) is between 5 and 21.
- (D) \(9\): Valid, since \(9\) is between 5 and 21.
- (E) \(15\): Valid, since \(15\) is between 5 and 21.
- (F) \(22\): Not valid, since \(22>21\).
So the correct possible lengths are \(6, 9, 15\).
Final Answer: \[ \boxed{6, \, 9, \, 15} \]