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In [tex]$\triangle ABC, AB = x, BC = y$[/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]$M, N$[/tex], and [tex]$O$[/tex] correspond to [tex]$A, B$[/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
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].

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