Answered

In [tex]\(\triangle ABC\)[/tex], [tex]\(AB = x\)[/tex], [tex]\(BC = y\)[/tex], and [tex]\(CA = 2x\)[/tex]. A similarity transformation with a scale factor of 0.5 maps [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle MNO\)[/tex], such that vertices [tex]\(M\)[/tex], [tex]\(N\)[/tex], and [tex]\(O\)[/tex] correspond to [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex], respectively. If [tex]\(OM = 5\)[/tex], what is [tex]\(AB\)[/tex]?

A. [tex]\(AB = 2.5\)[/tex]

B. [tex]\(AB = 10\)[/tex]

C. [tex]\(AB = 5\)[/tex]

D. [tex]\(AB = 1.25\)[/tex]

E. [tex]\(AB = 2\)[/tex]



Answer :

GinnyAnswer
Given the problem, let's solve it step by step using the conditions provided.

1. We know that a similarity transformation with a scale factor of 0.5 maps [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle MNO\)[/tex]. This means that each side of [tex]\(\triangle MNO\)[/tex] is 0.5 (or half) the length of the corresponding side in [tex]\(\triangle ABC\)[/tex].

2. The transformation maps [tex]\( \triangle ABC \)[/tex] to [tex]\( \triangle MNO \)[/tex] such that:
- [tex]\( M \leftrightarrow A \)[/tex]
- [tex]\( N \leftrightarrow B \)[/tex]
- [tex]\( O \leftrightarrow C \)[/tex]

3. It is given that [tex]\( OM = 5 \)[/tex]. Since [tex]\( O \leftrightarrow C \)[/tex] and [tex]\( M \leftrightarrow A \)[/tex], the length [tex]\( OM \)[/tex] corresponds to [tex]\( CA \)[/tex] in [tex]\(\triangle ABC\)[/tex].

4. Because of the similarity transformation, the length of [tex]\( CA \)[/tex] in [tex]\(\triangle ABC\)[/tex] is twice the length of [tex]\( OM \)[/tex]:
[tex]\[ CA = 2 \times OM = 2 \times 5 = 10 \][/tex]

5. It's given that [tex]\( CA = 2x \)[/tex]. By equating, we have:
[tex]\[ 2x = 10 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = 5 \][/tex]

6. The length [tex]\( AB = x \)[/tex]. Therefore, substituting the value we found:
[tex]\[ AB = 5 \][/tex]

Thus, the length of [tex]\( AB \)[/tex] is [tex]\( \boxed{5} \)[/tex]. So, the correct answer is:
[tex]\[ \text{C. } AB = 5 \][/tex]

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