To solve the problem, we need to use the properties of similar triangles and the given scale factor. Here are the detailed, step-by-step calculations:
1. Understand the Problem:
- We have a triangle [tex]\( \triangle ABC \)[/tex], where [tex]\( AB = x \)[/tex], [tex]\( BC = y \)[/tex], and [tex]\( CA = 2x \)[/tex].
- This triangle is transformed into a similar triangle [tex]\( \triangle MNO \)[/tex] with a scale factor of 0.5, such that [tex]\( M \)[/tex] corresponds to [tex]\( A \)[/tex], [tex]\( N \)[/tex] to [tex]\( B \)[/tex], and [tex]\( O \)[/tex] to [tex]\( C \)[/tex].
- We are given that [tex]\( OM = 5 \)[/tex] in the transformed triangle.
2. Identify Corresponding Sides:
- The side [tex]\( OM \)[/tex] in [tex]\( \triangle MNO \)[/tex] corresponds to [tex]\( CA \)[/tex] in [tex]\( \triangle ABC \)[/tex].
3. Relationship Between Corresponding Sides Using the Scale Factor:
- Since the similarity transformation has a scale factor of 0.5, the lengths of corresponding sides in the two triangles are related by this factor. Specifically, [tex]\( OM = 0.5 \times CA \)[/tex].
4. Given Value Substitution:
- We know [tex]\( OM = 5 \)[/tex], so using the scale factor:
[tex]\[
5 = 0.5 \times CA
\][/tex]
5. Solve for [tex]\( CA \)[/tex]:
[tex]\[
CA = \frac{5}{0.5} = 10
\][/tex]
6. Express [tex]\( CA \)[/tex] in Terms of [tex]\( x \)[/tex]:
- From the given problem statement, [tex]\( CA = 2x \)[/tex].
7. Set Up the Equation Using the Value Found for [tex]\( CA \)[/tex]:
[tex]\[
2x = 10
\][/tex]
8. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{10}{2} = 5
\][/tex]
9. Determine [tex]\( AB \)[/tex]:
- The side [tex]\( AB \)[/tex] is given by [tex]\( x \)[/tex] in [tex]\( \triangle ABC \)[/tex]. Thus,
[tex]\[
AB = x = 5
\][/tex]
Hence, the value of [tex]\( AB \)[/tex] is [tex]\( \boxed{5} \)[/tex].
