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What are the new coordinates of point A when it is rotated about the origin by

a) [tex]90^{\circ}[/tex] clockwise?
b) [tex]180^{\circ}[/tex]?
c) [tex]270^{\circ}[/tex] clockwise?

Sagot :

GinnyAnswer
Sure! Let's walk through the transformations step by step for the given point [tex]\( A(1, 1) \)[/tex].

### a) Rotation by [tex]\(90^\circ\)[/tex] clockwise:
For a point [tex]\((x, y)\)[/tex], the transformation formulas for a [tex]\(90^\circ\)[/tex] clockwise rotation are:
- [tex]\( x' = y \)[/tex]
- [tex]\( y' = -x \)[/tex]

Applying these formulas to the point [tex]\( A(1, 1) \)[/tex]:
- [tex]\( x' = 1 \)[/tex]
- [tex]\( y' = -1 \)[/tex]

Thus, the new coordinates after a [tex]\(90^\circ\)[/tex] clockwise rotation are [tex]\((1, -1)\)[/tex].

### b) Rotation by [tex]\(180^\circ\)[/tex]:
For a point [tex]\((x, y)\)[/tex], the transformation formulas for a [tex]\(180^\circ\)[/tex] rotation are:
- [tex]\( x' = -x \)[/tex]
- [tex]\( y' = -y \)[/tex]

Applying these formulas to the point [tex]\( A(1, 1) \)[/tex]:
- [tex]\( x' = -1 \)[/tex]
- [tex]\( y' = -1 \)[/tex]

Thus, the new coordinates after a [tex]\(180^\circ\)[/tex] rotation are [tex]\((-1, -1)\)[/tex].

### c) Rotation by [tex]\(270^\circ\)[/tex] clockwise:
For a point [tex]\((x, y)\)[/tex], the transformation formulas for a [tex]\(270^\circ\)[/tex] clockwise rotation are:
- [tex]\( x' = -y \)[/tex]
- [tex]\( y' = x \)[/tex]

Applying these formulas to the point [tex]\( A(1, 1) \)[/tex]:
- [tex]\( x' = -1 \)[/tex]
- [tex]\( y' = 1 \)[/tex]

Thus, the new coordinates after a [tex]\(270^\circ\)[/tex] clockwise rotation are [tex]\((-1, 1)\)[/tex].

In summary:
a) After a [tex]\(90^\circ\)[/tex] clockwise rotation, the coordinates are [tex]\((1, -1)\)[/tex].
b) After a [tex]\(180^\circ\)[/tex] rotation, the coordinates are [tex]\((-1, -1)\)[/tex].
c) After a [tex]\(270^\circ\)[/tex] clockwise rotation, the coordinates are [tex]\((-1, 1)\)[/tex].
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