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Find the coordinates of the image of a kite whose vertices are [tex]P(0,8)[/tex], [tex]Q(3,3)[/tex], [tex]R(0,6)[/tex], and [tex]S(-3,3)[/tex] rotated about the origin through:

a) [tex]90^{\circ}[/tex]
b) [tex]180^{\circ}[/tex]



Answer :

GinnyAnswer
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]

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