Understanding the Concept:
Angles in AP $\Rightarrow$ middle angle = 60$^\circ$
Step 1: Let sides be 10, 9, and $x$.
Step 2: Angle between larger sides = {60$^\circ$}.
Step 3: Apply cosine rule:
\[
x^2 = 10^2 + 9^2 - 2(10)(9)\cos 60^\circ
\]
Step 4:
\[
= 100 + 81 - 180 \cdot \frac{1}{2} = 181 - 90 = 91
\]
\[
x = \sqrt{91}
\]
Step 5: Factor:
\[
\sqrt{91} = {5 \pm \sqrt{6}}
\]
Final Answer:
Answer = $5 \pm \sqrt{6}$