Given that a similarity transformation with a scale factor of 0.5 maps [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle MNO\)[/tex] such that [tex]\(M\)[/tex], [tex]\(N\)[/tex], and [tex]\(O\)[/tex] correspond to [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex], respectively, we need to find the length of [tex]\(AB\)[/tex] in [tex]\(\triangle ABC\)[/tex] if [tex]\(OM = 5\)[/tex] in [tex]\(\triangle MNO\)[/tex].
1. Understanding similarity transformations:
- In a similarity transformation, corresponding sides of similar triangles are in the same ratio, which is given by the scale factor.
- If the scale factor is 0.5, then each side of [tex]\(\triangle MNO\)[/tex] is half the corresponding side of [tex]\(\triangle ABC\)[/tex].
2. Given information:
- [tex]\(OM = 5\)[/tex] in [tex]\(\triangle MNO\)[/tex].
3. Relate [tex]\(OM\)[/tex] back to [tex]\(\triangle ABC\)[/tex]:
- [tex]\(OM\)[/tex] corresponds to side [tex]\(AC\)[/tex] of [tex]\(\triangle ABC\)[/tex]. However, given [tex]\(OM = 5\)[/tex] and the scale factor 0.5, this means that before the transformation, the side [tex]\(AC\)[/tex] was twice as long as [tex]\(OM\)[/tex]. Thus, [tex]\(AC = OM / 0.5 = 5 / 0.5 = 10\)[/tex].
4. Conclusion:
- The corresponding length in [tex]\(\triangle ABC\)[/tex] before the similarity transformation is found by dividing the length in [tex]\(\triangle MNO\)[/tex] by the scale factor (i.e., 0.5), resulting in [tex]\(AB = OM / 0.5\)[/tex].
Given that we perform the calculation correctly:
[tex]\[
AB = \frac{OM}{\text{scale factor}} = \frac{5}{0.5} = 10
\][/tex]
Thus, the length of [tex]\(AB\)[/tex] is:
[tex]\[
\boxed{10}
\][/tex]
Therefore, the correct answer is:
B. [tex]\(AB = 10\)[/tex]
