Given the problem, let's solve it step by step using the conditions provided.
1. We know that a similarity transformation with a scale factor of 0.5 maps [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle MNO\)[/tex]. This means that each side of [tex]\(\triangle MNO\)[/tex] is 0.5 (or half) the length of the corresponding side in [tex]\(\triangle ABC\)[/tex].
2. The transformation maps [tex]\( \triangle ABC \)[/tex] to [tex]\( \triangle MNO \)[/tex] such that:
- [tex]\( M \leftrightarrow A \)[/tex]
- [tex]\( N \leftrightarrow B \)[/tex]
- [tex]\( O \leftrightarrow C \)[/tex]
3. It is given that [tex]\( OM = 5 \)[/tex]. Since [tex]\( O \leftrightarrow C \)[/tex] and [tex]\( M \leftrightarrow A \)[/tex], the length [tex]\( OM \)[/tex] corresponds to [tex]\( CA \)[/tex] in [tex]\(\triangle ABC\)[/tex].
4. Because of the similarity transformation, the length of [tex]\( CA \)[/tex] in [tex]\(\triangle ABC\)[/tex] is twice the length of [tex]\( OM \)[/tex]:
[tex]\[
CA = 2 \times OM = 2 \times 5 = 10
\][/tex]
5. It's given that [tex]\( CA = 2x \)[/tex]. By equating, we have:
[tex]\[
2x = 10
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = 5
\][/tex]
6. The length [tex]\( AB = x \)[/tex]. Therefore, substituting the value we found:
[tex]\[
AB = 5
\][/tex]
Thus, the length of [tex]\( AB \)[/tex] is [tex]\( \boxed{5} \)[/tex]. So, the correct answer is:
[tex]\[
\text{C. } AB = 5
\][/tex]
