Question

one of the vertices of the original rectangle has the coordinates
$(6,3)$.
after a rotation of $90^{circ}$, the coordinates of this vertex would be

one of the vertices of the original rectangle has the coordinates
$(0,3)$.
after a rotation of $180^{circ}$, the coordinates of this vertex would be

Explanation:

Step1: Recall rotation rules

For a \(90^{\circ}\) rotation counter - clockwise about the origin \((x,y)\to(-y,x)\). For a \(180^{\circ}\) rotation about the origin \((x,y)\to(-x,-y)\).

Step2: Apply \(90^{\circ}\) rotation rule

Given the point \((6,3)\). Using the \(90^{\circ}\) rotation formula \((x = 6,y = 3)\), we get \((-y,x)=(- 3,6)\).

Step3: Apply \(180^{\circ}\) rotation rule

Given the point \((0,3)\). Using the \(180^{\circ}\) rotation formula \((x = 0,y = 3)\), we get \((-x,-y)=(0,-3)\)

Answer:

For the \(90^{\circ}\) rotation: \((-3,6)\)
For the \(180^{\circ}\) rotation: \((0,-3)\)