Question

If 4 and 11 are the lengths of two sides of a triangular region, which of the following can be the length of the third side? I. 5
II. 13
III. 15

• A.
• B.
• C.
• D.
• E.

Hint:

The triangle inequality theorem is key when determining possible lengths of a triangle's sides.

Answer 1

The Correct Option is C

Step 1: Apply the triangle inequality theorem.
The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For sides \( a \), \( b \), and \( c \), the following inequalities must hold: \[ a + b>c, \quad a + c>b, \quad b + c>a. \] In our case, the known sides are 4 and 11. Let the third side be \( x \). The inequalities are: \[ 4 + 11>x \quad \text{or} \quad x<15, \] \[ 4 + x>11 \quad \text{or} \quad x>7, \] \[ 11 + x>4 \quad \text{which is always true since \( x>0 \)}. \] Thus, the third side \( x \) must satisfy: \[ 7<x<15. \] Step 2: Analyze the options.
- (I) \( x = 5 \): This does not satisfy \( x>7 \), so 5 cannot be the third side.
- (II) \( x = 13 \): This satisfies \( 7<x<15 \), so 13 can be the third side.
- (III) \( x = 15 \): This does not satisfy \( x<15 \), so 15 cannot be the third side. Conclusion:
Thus, the possible third sides are 13, so the correct answer is (C) I and II only.