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]$AB = 125$[/tex]

E. [tex]$AB = 2$[/tex]



Answer :

GinnyAnswer
To solve this problem, we need to understand the relationships between the sides of the two triangles and the scale factor given by the similarity transformation.

Given:
- [tex]\(\bigtriangleup ABC\)[/tex] is mapped to [tex]\(\bigtriangleup MNO\)[/tex] with a scale factor of 0.5.
- The corresponding vertices are such that [tex]\( M \)[/tex] corresponds to [tex]\( A \)[/tex], [tex]\( N \)[/tex] corresponds to [tex]\( B \)[/tex], and [tex]\( O \)[/tex] corresponds to [tex]\( C \)[/tex].
- We are given [tex]\( OM = 5 \)[/tex].
- In [tex]\(\bigtriangleup ABC\)[/tex], we know [tex]\( CA = 2x \)[/tex] and we need to determine the length of [tex]\( AB \)[/tex].

Step 1: Understand the effect of the scale factor.

Since the triangles are similar by a scale factor of 0.5, every segment in [tex]\(\bigtriangleup ABC\)[/tex] is scaled down by this factor to obtain [tex]\(\bigtriangleup MNO\)[/tex]. Thus:
[tex]\[ OM \text{ (corresponding to } CA ) = 0.5 \times CA \][/tex]

Step 2: Relate the given length [tex]\( OM \)[/tex] to [tex]\( CA \)[/tex] in [tex]\(\bigtriangleup ABC\)[/tex].

Since [tex]\( OM = 5 \)[/tex]:
[tex]\[ 5 = 0.5 \times CA \][/tex]

Step 3: Solve for [tex]\( CA \)[/tex]:

[tex]\[ CA = \frac{5}{0.5} \][/tex]
[tex]\[ CA = 10 \][/tex]

Step 4: Use the relationship [tex]\( CA = 2x \)[/tex] to find [tex]\( x \)[/tex]:

[tex]\[ 2x = 10 \][/tex]
[tex]\[ x = \frac{10}{2} \][/tex]
[tex]\[ x = 5 \][/tex]

Step 5: Determine [tex]\( AB \)[/tex] in [tex]\(\bigtriangleup ABC\)[/tex]:

By the problem setup, [tex]\( AB = x \)[/tex]. Hence:
[tex]\[ AB = 5 \][/tex]

Thus, the length of [tex]\( AB \)[/tex] is:
[tex]\[ \boxed{5} \][/tex]