To solve the problem, let's follow the geometric transformations step by step. We're given the coordinates of triangle [tex]\( PQR \)[/tex], which are [tex]\( P(1,2) \)[/tex], [tex]\( Q(3,3) \)[/tex], and [tex]\( R(2,4) \)[/tex]. We'll apply the translation and reflection to these points to find the coordinates of triangle [tex]\( XYZ \)[/tex].
1. Translation: Translate 2 units to the right and 1 unit down
* For point [tex]\( P(1, 2) \)[/tex]:
[tex]\[
P' = \left( 1 + 2, 2 - 1 \right) = (3, 1)
\][/tex]
* For point [tex]\( Q(3, 3) \)[/tex]:
[tex]\[
Q' = \left( 3 + 2, 3 - 1 \right) = (5, 2)
\][/tex]
* For point [tex]\( R(2, 4) \)[/tex]:
[tex]\[
R' = \left( 2 + 2, 4 - 1 \right) = (4, 3)
\][/tex]
After translation, the new coordinates are [tex]\( P'(3,1) \)[/tex], [tex]\( Q'(5,2) \)[/tex], and [tex]\( R'(4,3) \)[/tex].
2. Reflection: Reflect across the [tex]\( x \)[/tex]-axis (change the sign of the [tex]\( y \)[/tex]-coordinate)
* For point [tex]\( P'(3, 1) \)[/tex]:
[tex]\[
X = (3, -1)
\][/tex]
* For point [tex]\( Q'(5, 2) \)[/tex]:
[tex]\[
Y = (5, -2)
\][/tex]
* For point [tex]\( R'(4, 3) \)[/tex]:
[tex]\[
Z = (4, -3)
\][/tex]
After reflection across the [tex]\( x \)[/tex]-axis, the final coordinates are [tex]\( X(3, -1) \)[/tex], [tex]\( Y(5, -2) \)[/tex], and [tex]\( Z(4, -3) \)[/tex].
Thus, the coordinates of the vertices of triangle [tex]\( XYZ \)[/tex] are:
[tex]\[
\boxed{X(3, -1), Y(5, -2), and Z(4, -3)}
\][/tex]
This matches option C.
