At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
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.
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.
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.