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]
