Answer :
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]
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]