If triangle [tex]$XYZ$[/tex] is translated using the rule [tex]$(x, y) \rightarrow (x+5, y-3)$[/tex] and then rotated [tex][tex]$90^{\circ}$[/tex][/tex] clockwise to create triangle [tex]$X'Y'Z'$[/tex], what is the location of [tex]$X'$[/tex]?

A. [tex]$(-3, 10)$[/tex]
B. [tex][tex]$(-2, -15)$[/tex][/tex]
C. [tex]$(-2, -7)$[/tex]
D. [tex]$(3, -10)$[/tex]



Answer :

Let's analyze the given transformations step-by-step for the point [tex]\(X\)[/tex] with coordinates [tex]\((-3, 10)\)[/tex].

### Step 1: Translation
The translation rule given is [tex]\((x, y) \rightarrow (x+5, y-3)\)[/tex].

Applying the translation to [tex]\(X\)[/tex]:
[tex]\[ x' = -3 + 5 = 2 \][/tex]
[tex]\[ y' = 10 - 3 = 7 \][/tex]

So, after the translation, the coordinates of [tex]\(X\)[/tex] are [tex]\((2, 7)\)[/tex].

### Step 2: Rotation
The rotation rule for a [tex]\(90^\circ\)[/tex] clockwise rotation is [tex]\((x, y) \rightarrow (y, -x)\)[/tex].

Applying the rotation to the translated coordinates [tex]\((2, 7)\)[/tex]:
[tex]\[ x'' = y' = 7 \][/tex]
[tex]\[ y'' = -x' = -2 \][/tex]

So, after the rotation, the coordinates of [tex]\(X\)[/tex] are [tex]\((7, -2)\)[/tex].

### Conclusion
The location of [tex]\(X'\)[/tex] after translating the point [tex]\((-3, 10)\)[/tex] using the rule [tex]\((x, y) \rightarrow (x+5, y-3)\)[/tex] and then rotating the point [tex]\(90^\circ\)[/tex] clockwise is [tex]\((7, -2)\)[/tex].

Thus, the correct answer is not listed among the provided options exactly, but analyzing the given details, the closest matching step appears to be:
[tex]\[ (7, -2) \][/tex]
Which is not in the provided options—seems like there is a discrepancy or error in problem statement options.

Therefore, please reach out for further clarification or correction in the options provided.