To solve the problem, let's analyze the given information step-by-step meticulously:
1. Similarity Transformation Concept: We know that a similarity transformation keeps the shapes of triangles the same but changes their sizes by a specific ratio, known as the scale factor.
2. Scale Factor: The problem states that a similarity transformation with a scale factor of [tex]\(0.5\)[/tex] (or [tex]\(\frac{1}{2}\)[/tex]) is applied to [tex]\(\triangle ABC\)[/tex], resulting in [tex]\(\triangle MNO\)[/tex], with vertices [tex]\(M\)[/tex], [tex]\(N\)[/tex], and [tex]\(O\)[/tex] corresponding to [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] respectively.
3. Length of [tex]\(OM\)[/tex]: We are provided with [tex]\(OM = 5\)[/tex], which is the image of [tex]\(CA\)[/tex] after the transformation.
4. Relationship between Original and Transformed Lengths:
[tex]\[
OM = \text{scale factor} \times CA \quad \text{or} \quad 5 = 0.5 \times CA
\][/tex]
5. Solving for [tex]\(CA\)[/tex]: To find [tex]\(CA\)[/tex], we solve the above equation:
[tex]\[
5 = 0.5 \times CA
\][/tex]
[tex]\[
CA = \frac{5}{0.5} = 10
\][/tex]
6. Given Relationship [tex]\(CA = 2x\)[/tex]: In the original [tex]\(\triangle ABC\)[/tex], it is given that [tex]\(CA = 2x\)[/tex]. We already found that [tex]\(CA = 10\)[/tex].
[tex]\[
2x = 10
\][/tex]
7. Solving for [tex]\(x\)[/tex]: We now solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{10}{2} = 5
\][/tex]
8. Length of [tex]\(AB\)[/tex]: Recall from the given notation that [tex]\(AB = x\)[/tex]. Therefore, substituting [tex]\(x\)[/tex],
[tex]\[
AB = 5
\][/tex]
Thus, the correct answer is [tex]\(C. \, AB = 5\)[/tex].
