To solve this problem, let’s analyze the given information step-by-step.
First, we know that there is a similarity transformation with a scale factor of 0.5 mapping [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle MNO\)[/tex], and according to the problem statement, the corresponding vertices have the following relationships:
- [tex]\(M\)[/tex] corresponds to [tex]\(A\)[/tex]
- [tex]\(N\)[/tex] corresponds to [tex]\(B\)[/tex]
- [tex]\(O\)[/tex] corresponds to [tex]\(C\)[/tex]
Given that [tex]\(OM = 5\)[/tex] in the transformed triangle [tex]\(\triangle MNO\)[/tex], we need to find the original side length [tex]\(AB\)[/tex] in [tex]\(\triangle ABC\)[/tex].
1. Understanding Similarity Transformation:
A similarity transformation with a scale factor of 0.5 means that each side length in [tex]\(\triangle MNO\)[/tex] is half of the corresponding side length in [tex]\(\triangle ABC\)[/tex].
2. Calculating the Corresponding Side:
Given [tex]\(OM = 5\)[/tex] in [tex]\(\triangle MNO\)[/tex], we need to find what the original length of the side [tex]\(AB\)[/tex] was in [tex]\(\triangle ABC\)[/tex].
Since [tex]\(OM\)[/tex] in [tex]\(\triangle MNO\)[/tex] corresponds to [tex]\(CA\)[/tex] in [tex]\(\triangle ABC\)[/tex] and we know the scale factor is 0.5, we need to set up the proportion:
[tex]\[
OM = \frac{CA}{2}
\][/tex]
Substitute the value of [tex]\(OM\)[/tex]:
[tex]\[
5 = \frac{CA}{2}
\][/tex]
Solving for [tex]\(CA\)[/tex]:
[tex]\[
CA = 5 \times 2 = 10
\][/tex]
3. Finding [tex]\(AB\)[/tex]:
Knowing [tex]\(CA = 2x\)[/tex] as given in the problem statement, we can now determine the value of [tex]\(x\)[/tex]:
[tex]\[
2x = 10 \implies x = 5
\][/tex]
Hence, [tex]\(AB = x = 5\)[/tex].
Therefore, the length of [tex]\(AB\)[/tex] in [tex]\(\triangle ABC\)[/tex] is:
[tex]\[
AB = 10
\][/tex]
The correct answer is B. [tex]\(AB = 10\)[/tex].
