Answer :
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].
### 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].
