Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
Certainly! Let's find the coordinates of the vertices of the kite rotated about the origin through 90° and 180°.
### Rotation by 90°
1. Rotating Point P (0, 8) by 90°:
- The formula for rotating a point \((x, y)\) by 90° counterclockwise around the origin is:
[tex]\[ (x', y') = (-y, x) \][/tex]
- For point \(P(0, 8)\):
[tex]\[ x' = -8, \quad y' = 0 \][/tex]
- New coordinates for \(P\) are:
[tex]\[ P'(0, 8) = (-8, 0) \][/tex]
2. Rotating Point Q (3, 3) by 90°:
- For point \(Q(3, 3)\):
[tex]\[ x' = -3, \quad y' = 3 \][/tex]
- New coordinates for \(Q\) are:
[tex]\[ Q(3, 3) = (-3, 3) \][/tex]
3. Rotating Point R (0, 6) by 90°:
- For point \(R(0, 6)\):
[tex]\[ x' = -6, \quad y' = 0 \][/tex]
- New coordinates for \(R\) are:
[tex]\[ R(0, 6) = (-6, 0) \][/tex]
4. Rotating Point S (-3, 3) by 90°:
- For point \(S(-3, 3)\):
[tex]\[ x' = -3, \quad y' = -3 \][/tex]
- New coordinates for \(S\) are:
[tex]\[ S(-3, 3) = (-3, -3) \][/tex]
So, the coordinates of the vertices of the kite after a 90° rotation are:
[tex]\[ \{P'(-8, 0), Q'(-3, 3), R'(-6, 0), S'(-3, -3)\} \][/tex]
### Rotation by 180°
1. Rotating Point P (0, 8) by 180°:
- The formula for rotating a point \((x, y)\) by 180° around the origin is:
[tex]\[ (x', y') = (-x, -y) \][/tex]
- For point \(P(0, 8)\):
[tex]\[ x' = 0, \quad y' = -8 \][/tex]
- New coordinates for \(P\) are:
[tex]\[ P(0, 8) = (0, -8) \][/tex]
2. Rotating Point Q (3, 3) by 180°:
- For point \(Q(3, 3)\):
[tex]\[ x' = -3, \quad y' = -3 \][/tex]
- New coordinates for \(Q\) are:
[tex]\[ Q(3, 3) = (-3, -3) \][/tex]
3. Rotating Point R (0, 6) by 180°:
- For point \(R(0, 6)\):
[tex]\[ x' = 0, \quad y' = -6 \][/tex]
- New coordinates for \(R\) are:
[tex]\[ R(0, 6) = (0, -6) \][/tex]
4. Rotating Point S (-3, 3) by 180°:
- For point \(S(-3, 3)\):
[tex]\[ x' = 3, \quad y' = -3 \][/tex]
- New coordinates for \(S\) are:
[tex]\[ S(-3, 3) = (3, -3) \][/tex]
So, the coordinates of the vertices of the kite after a 180° rotation are:
[tex]\[ \{P'(0, -8), Q'(-3, -3), R'(0, -6), S'(3, -3)\} \][/tex]
In summary, the rotated coordinates of the kite are:
- After 90° rotation: \(\{P'(-8, 0), Q'(-3, 3), R'(-6, 0), S'(-3, -3)\}\)
- After 180° rotation: [tex]\(\{P'(0, -8), Q'(-3, -3), R'(0, -6), S'(3, -3)\}\)[/tex]
### Rotation by 90°
1. Rotating Point P (0, 8) by 90°:
- The formula for rotating a point \((x, y)\) by 90° counterclockwise around the origin is:
[tex]\[ (x', y') = (-y, x) \][/tex]
- For point \(P(0, 8)\):
[tex]\[ x' = -8, \quad y' = 0 \][/tex]
- New coordinates for \(P\) are:
[tex]\[ P'(0, 8) = (-8, 0) \][/tex]
2. Rotating Point Q (3, 3) by 90°:
- For point \(Q(3, 3)\):
[tex]\[ x' = -3, \quad y' = 3 \][/tex]
- New coordinates for \(Q\) are:
[tex]\[ Q(3, 3) = (-3, 3) \][/tex]
3. Rotating Point R (0, 6) by 90°:
- For point \(R(0, 6)\):
[tex]\[ x' = -6, \quad y' = 0 \][/tex]
- New coordinates for \(R\) are:
[tex]\[ R(0, 6) = (-6, 0) \][/tex]
4. Rotating Point S (-3, 3) by 90°:
- For point \(S(-3, 3)\):
[tex]\[ x' = -3, \quad y' = -3 \][/tex]
- New coordinates for \(S\) are:
[tex]\[ S(-3, 3) = (-3, -3) \][/tex]
So, the coordinates of the vertices of the kite after a 90° rotation are:
[tex]\[ \{P'(-8, 0), Q'(-3, 3), R'(-6, 0), S'(-3, -3)\} \][/tex]
### Rotation by 180°
1. Rotating Point P (0, 8) by 180°:
- The formula for rotating a point \((x, y)\) by 180° around the origin is:
[tex]\[ (x', y') = (-x, -y) \][/tex]
- For point \(P(0, 8)\):
[tex]\[ x' = 0, \quad y' = -8 \][/tex]
- New coordinates for \(P\) are:
[tex]\[ P(0, 8) = (0, -8) \][/tex]
2. Rotating Point Q (3, 3) by 180°:
- For point \(Q(3, 3)\):
[tex]\[ x' = -3, \quad y' = -3 \][/tex]
- New coordinates for \(Q\) are:
[tex]\[ Q(3, 3) = (-3, -3) \][/tex]
3. Rotating Point R (0, 6) by 180°:
- For point \(R(0, 6)\):
[tex]\[ x' = 0, \quad y' = -6 \][/tex]
- New coordinates for \(R\) are:
[tex]\[ R(0, 6) = (0, -6) \][/tex]
4. Rotating Point S (-3, 3) by 180°:
- For point \(S(-3, 3)\):
[tex]\[ x' = 3, \quad y' = -3 \][/tex]
- New coordinates for \(S\) are:
[tex]\[ S(-3, 3) = (3, -3) \][/tex]
So, the coordinates of the vertices of the kite after a 180° rotation are:
[tex]\[ \{P'(0, -8), Q'(-3, -3), R'(0, -6), S'(3, -3)\} \][/tex]
In summary, the rotated coordinates of the kite are:
- After 90° rotation: \(\{P'(-8, 0), Q'(-3, 3), R'(-6, 0), S'(-3, -3)\}\)
- After 180° rotation: [tex]\(\{P'(0, -8), Q'(-3, -3), R'(0, -6), S'(3, -3)\}\)[/tex]
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
