Question

determine the range of side ab.

Explanation:

Step1: Apply triangle - inequality theorem in left - hand triangle

In the triangle with sides 13, 22 and a side related to \(x\). According to 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. So \(|22 - 13|\lt\) the side related to \(x\lt22 + 13\), which simplifies to \(9\lt\) the side related to \(x\lt35\).

Step2: Apply triangle - inequality theorem in right - hand triangle

In the triangle with sides 29, 39 and the same side related to \(x\). We have \(|39 - 29|\lt x\lt39+29\), which simplifies to \(10\lt x\lt68\).

Step3: Find the intersection of the two ranges

To satisfy both triangle - inequality conditions simultaneously, we take the intersection of the two ranges. The lower bound is the maximum of 9 and 10, which is 10. The upper - bound is the minimum of 35 and 68, which is 35.

Answer:

\(10\lt x\lt35\)