Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Join our Q&A platform to get precise answers from experts in diverse fields and enhance your understanding. Get quick and reliable solutions to your questions from a community of experienced experts 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]
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.