To solve this problem, let's go through the steps one by one:
1. Identify Given Information:
- We are given that a similarity transformation maps [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle MNO\)[/tex] with a scale factor of 0.5.
- In [tex]\(\triangle MNO\)[/tex], [tex]\(OM = 5\)[/tex].
- The corresponding vertices are [tex]\(M \leftrightarrow A\)[/tex], [tex]\(N \leftrightarrow B\)[/tex], and [tex]\(O \leftrightarrow C\)[/tex].
- In [tex]\(\triangle ABC\)[/tex], [tex]\(CA = 2x\)[/tex], and we need to find [tex]\(x\)[/tex] (which corresponds to [tex]\(AB\)[/tex]).
2. Use the Scale Factor Relationship:
- The similarity transformation means that each side of [tex]\(\triangle MNO\)[/tex] is 0.5 (or half) of the respective side in [tex]\(\triangle ABC\)[/tex].
- [tex]\(OM\)[/tex] in [tex]\(\triangle MNO\)[/tex] corresponds to [tex]\(CA\)[/tex] in [tex]\(\triangle ABC\)[/tex].
3. Relate the Corresponding Sides:
- Given [tex]\(OM = 5\)[/tex] and knowing the scale factor from [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle MNO\)[/tex] is 0.5, we can set up the following relationship:
[tex]\[
OM = 0.5 \times CA
\][/tex]
- Substituting the given value of [tex]\(OM\)[/tex]:
[tex]\[
5 = 0.5 \times CA
\][/tex]
- Solving for [tex]\(CA\)[/tex]:
[tex]\[
CA = \frac{5}{0.5} = 10
\][/tex]
4. Determine [tex]\(AB\)[/tex]:
- We know [tex]\(CA = 2x\)[/tex] from [tex]\(\triangle ABC\)[/tex]:
[tex]\[
2x = CA
\][/tex]
- Since [tex]\(CA = 10\)[/tex]:
[tex]\[
2x = 10
\][/tex]
- Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{10}{2} = 5
\][/tex]
Thus, [tex]\(AB = x = 5\)[/tex].
So, the correct answer is:
[tex]\[ C. \, AB = 5 \][/tex]
