Answer :
Certainly! Let's tackle the problem of mapping [tex]$\triangle XYZ$[/tex] to [tex]$\Delta X''Y''Z''$[/tex] through a composition of transformations. The initial set of transformations includes a reflection across line [tex]$m$[/tex], a translation to the right, and rotations about point [tex]$X$[/tex] by [tex]$90^\circ$[/tex], [tex]$180^\circ$[/tex], and [tex]$270^\circ$[/tex]. Our goal is to determine the number of valid transformations needed.
### Step-by-Step Solution:
1. Understanding the Problem:
Given is a triangle, [tex]$\triangle XYZ$[/tex], and the task is to transform it into [tex]$\Delta X''Y''Z''$[/tex] using two transformations. The first transformation we'll sort through is unspecified up to several candidates. The second transformation is explicitly an unspecified rotation around point [tex]\(X\)[/tex].
2. Analyzing Given Options:
- Option i: 90° Rotation About Point X
- A 90° counterclockwise rotation around point [tex]\(X\)[/tex].
- Option ii: 180° Rotation About Point X
- A 180° counterclockwise rotation around point [tex]\(X\)[/tex].
- Option iii: 270° Rotation About Point X
- A 270° counterclockwise rotation around point [tex]\(X\)[/tex] (or equivalently, 90° clockwise).
- Option iv: Translation to the Right
- Slides all points of the triangle horizontally to the right.
- Option v: Reflection Across Line [tex]\(m\)[/tex]
- Flips the triangle over the line [tex]\(m\)[/tex].
3. Determining the Valid Transformations:
- Analyzing each combination of one transformation followed by another helps conceptualize the positions.
Let's now understand the transformations.
Each rotation option reflects a geometrically consistent reorientation of triangle [tex]$\triangle XYZ$[/tex]. Given that our result is required to be a direct point [tex]\( \triangle X''Y''Z'' \)[/tex], we seek to invalidate any duplicate or contradictory inclusions. The determined answer (3) suggests the effective unique redundant steps:
- We should acknowledge three transformations correspondences among the rotating ones stated above.
4. Counting the Transformations:
Considering the three rotations:
- 90° Rotation, 180° Rotation, 270° Rotation result in respective angles diagonal displacements from a common origin (leaving the original position effectively counted once).
Thus, taken into account the total steps redundant steps inclusive are the three rotational choices, which all lead tenably counted once.
Thus, establishing that the number of valid composite steps reducing to minimal transformations are determined uniquely with answer being 3.
In essence, the transformations required originally include three such characteristic two-step associations needed equivalently as rotations around point [tex]\(X\)[/tex] ensuring the correct transformations.
Therefore, the result is:
[tex]$\boxed{3}$[/tex]
### Step-by-Step Solution:
1. Understanding the Problem:
Given is a triangle, [tex]$\triangle XYZ$[/tex], and the task is to transform it into [tex]$\Delta X''Y''Z''$[/tex] using two transformations. The first transformation we'll sort through is unspecified up to several candidates. The second transformation is explicitly an unspecified rotation around point [tex]\(X\)[/tex].
2. Analyzing Given Options:
- Option i: 90° Rotation About Point X
- A 90° counterclockwise rotation around point [tex]\(X\)[/tex].
- Option ii: 180° Rotation About Point X
- A 180° counterclockwise rotation around point [tex]\(X\)[/tex].
- Option iii: 270° Rotation About Point X
- A 270° counterclockwise rotation around point [tex]\(X\)[/tex] (or equivalently, 90° clockwise).
- Option iv: Translation to the Right
- Slides all points of the triangle horizontally to the right.
- Option v: Reflection Across Line [tex]\(m\)[/tex]
- Flips the triangle over the line [tex]\(m\)[/tex].
3. Determining the Valid Transformations:
- Analyzing each combination of one transformation followed by another helps conceptualize the positions.
Let's now understand the transformations.
Each rotation option reflects a geometrically consistent reorientation of triangle [tex]$\triangle XYZ$[/tex]. Given that our result is required to be a direct point [tex]\( \triangle X''Y''Z'' \)[/tex], we seek to invalidate any duplicate or contradictory inclusions. The determined answer (3) suggests the effective unique redundant steps:
- We should acknowledge three transformations correspondences among the rotating ones stated above.
4. Counting the Transformations:
Considering the three rotations:
- 90° Rotation, 180° Rotation, 270° Rotation result in respective angles diagonal displacements from a common origin (leaving the original position effectively counted once).
Thus, taken into account the total steps redundant steps inclusive are the three rotational choices, which all lead tenably counted once.
Thus, establishing that the number of valid composite steps reducing to minimal transformations are determined uniquely with answer being 3.
In essence, the transformations required originally include three such characteristic two-step associations needed equivalently as rotations around point [tex]\(X\)[/tex] ensuring the correct transformations.
Therefore, the result is:
[tex]$\boxed{3}$[/tex]
