Question

if two sides of a triangle have lengths 4 and 9, then the length of the third side may be any number
greater than 4 but less than 9
less than 13
greater than 5
greater than 5 but less than 13

Explanation:

Step1: Recall the triangle inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Let the third side be \(x\).

Step2: Apply the first part of the theorem

\(x + 4>9\), which simplifies to \(x>9 - 4\), so \(x>5\).

Step3: Apply the second part of the theorem

\(x+9 > 4\) (which is always true for positive \(x\)).

Step4: Apply the third part of the theorem

\(4 + 9>x\), which simplifies to \(x<13\).

Answer:

D. greater than 5 but less than 13