To solve the problem, let's break it down into clear, step-by-step processes.
1. Understanding the Problem:
- We have an original triangle [tex]\( \triangle ABC \)[/tex] with sides [tex]\( AB = x \)[/tex], [tex]\( BC = y \)[/tex], and [tex]\( CA = 2x \)[/tex].
- This triangle undergoes a similarity transformation with a scale factor of 0.5, mapping it to a new triangle [tex]\( \triangle MNO \)[/tex], where vertices [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex] map to [tex]\( M, N, \)[/tex] and [tex]\( O \)[/tex] respectively.
- We are given that [tex]\( OM = 5 \)[/tex] and need to find the length of [tex]\( AB \)[/tex].
2. Using the Scale Factor:
- In the transformed triangle [tex]\( \triangle MNO \)[/tex], all sides are half the length of the sides in [tex]\( \triangle ABC \)[/tex].
- Since [tex]\( OM \)[/tex] corresponds to [tex]\( CA \)[/tex] and [tex]\( OM = 5 \)[/tex]:
[tex]\[
OM = \text{Scale Factor} \times CA
\][/tex]
[tex]\[
5 = 0.5 \times CA
\][/tex]
- Solving for [tex]\( CA \)[/tex]:
[tex]\[
CA = \frac{5}{0.5} = 10
\][/tex]
3. Finding [tex]\( x \)[/tex]:
- We know that [tex]\( CA = 2x \)[/tex]:
[tex]\[
2x = 10
\][/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{10}{2} = 5
\][/tex]
4. Conclusion:
- Since [tex]\( AB = x \)[/tex], we found that:
[tex]\[
AB = 5
\][/tex]
The correct answer is [tex]\( AB = 5 \)[/tex].
Therefore, the answer is [tex]\( \boxed{5} \)[/tex].
