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]$M, N$[/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 the problem, let's break it down into clear, step-by-step processes.

1. Understanding the Problem:
- We have an original triangle [tex]\( \triangle ABC \)[/tex] with sides [tex]\( AB = x \)[/tex], [tex]\( BC = y \)[/tex], and [tex]\( CA = 2x \)[/tex].
- This triangle undergoes a similarity transformation with a scale factor of 0.5, mapping it to a new triangle [tex]\( \triangle MNO \)[/tex], where vertices [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex] map to [tex]\( M, N, \)[/tex] and [tex]\( O \)[/tex] respectively.
- We are given that [tex]\( OM = 5 \)[/tex] and need to find the length of [tex]\( AB \)[/tex].

2. Using the Scale Factor:
- In the transformed triangle [tex]\( \triangle MNO \)[/tex], all sides are half the length of the sides in [tex]\( \triangle ABC \)[/tex].
- Since [tex]\( OM \)[/tex] corresponds to [tex]\( CA \)[/tex] and [tex]\( OM = 5 \)[/tex]:
[tex]\[ OM = \text{Scale Factor} \times CA \][/tex]
[tex]\[ 5 = 0.5 \times CA \][/tex]
- Solving for [tex]\( CA \)[/tex]:
[tex]\[ CA = \frac{5}{0.5} = 10 \][/tex]

3. Finding [tex]\( x \)[/tex]:
- We know that [tex]\( CA = 2x \)[/tex]:
[tex]\[ 2x = 10 \][/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{10}{2} = 5 \][/tex]

4. Conclusion:
- Since [tex]\( AB = x \)[/tex], we found that:
[tex]\[ AB = 5 \][/tex]

The correct answer is [tex]\( AB = 5 \)[/tex].

Therefore, the answer is [tex]\( \boxed{5} \)[/tex].

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