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 :

To solve the problem, we need to use the properties of similar triangles and the given scale factor. Here are the detailed, step-by-step calculations:

1. Understand the Problem:
- We have a triangle [tex]\( \triangle ABC \)[/tex], where [tex]\( AB = x \)[/tex], [tex]\( BC = y \)[/tex], and [tex]\( CA = 2x \)[/tex].
- This triangle is transformed into a similar triangle [tex]\( \triangle MNO \)[/tex] with a scale factor of 0.5, such that [tex]\( M \)[/tex] corresponds to [tex]\( A \)[/tex], [tex]\( N \)[/tex] to [tex]\( B \)[/tex], and [tex]\( O \)[/tex] to [tex]\( C \)[/tex].
- We are given that [tex]\( OM = 5 \)[/tex] in the transformed triangle.

2. Identify Corresponding Sides:
- The side [tex]\( OM \)[/tex] in [tex]\( \triangle MNO \)[/tex] corresponds to [tex]\( CA \)[/tex] in [tex]\( \triangle ABC \)[/tex].

3. Relationship Between Corresponding Sides Using the Scale Factor:
- Since the similarity transformation has a scale factor of 0.5, the lengths of corresponding sides in the two triangles are related by this factor. Specifically, [tex]\( OM = 0.5 \times CA \)[/tex].

4. Given Value Substitution:
- We know [tex]\( OM = 5 \)[/tex], so using the scale factor:
[tex]\[ 5 = 0.5 \times CA \][/tex]

5. Solve for [tex]\( CA \)[/tex]:
[tex]\[ CA = \frac{5}{0.5} = 10 \][/tex]

6. Express [tex]\( CA \)[/tex] in Terms of [tex]\( x \)[/tex]:
- From the given problem statement, [tex]\( CA = 2x \)[/tex].

7. Set Up the Equation Using the Value Found for [tex]\( CA \)[/tex]:
[tex]\[ 2x = 10 \][/tex]

8. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{10}{2} = 5 \][/tex]

9. Determine [tex]\( AB \)[/tex]:
- The side [tex]\( AB \)[/tex] is given by [tex]\( x \)[/tex] in [tex]\( \triangle ABC \)[/tex]. Thus,
[tex]\[ AB = x = 5 \][/tex]

Hence, the value of [tex]\( AB \)[/tex] is [tex]\( \boxed{5} \)[/tex].