Question

find the coordinates of the vertices of each figure after the given transformation. 1) rotation 90° counterclockwise about the origin 2) rotation 180° about the origin 3) rotation 90° counterclockwise about the origin 4) rotation 180° about the origin

Explanation:

1. For a \(90^{\circ}\) counter - clockwise rotation about the origin:

• The rule for a \(90^{\circ}\) counter - clockwise rotation of a point \((x,y)\) about the origin is \((x,y)\to(-y,x)\).
• Let's assume a point \(A=(x_1,y_1)\) in the original figure. After the \(90^{\circ}\) counter - clockwise rotation, its new coordinates \(A'=(-y_1,x_1)\).
• For example, if we have a point \((2,3)\), after a \(90^{\circ}\) counter - clockwise rotation about the origin, the new point is \((-3,2)\).

1. For a \(180^{\circ}\) rotation about the origin:

• The rule for a \(180^{\circ}\) rotation of a point \((x,y)\) about the origin is \((x,y)\to(-x,-y)\).
• Let \(B=(x_2,y_2)\) be a point in the original figure. After a \(180^{\circ}\) rotation, its new coordinates \(B'=(-x_2,-y_2)\). For instance, if we have a point \((4, - 1)\), after a \(180^{\circ}\) rotation about the origin, the new point is \((-4,1)\).

Since we don't have the specific coordinates of the vertices of the figures in the image, we can't give the exact numerical answers. But the general steps for finding the new coordinates after rotation are as above.

Step1: Recall rotation rules

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

Step2: Apply rules to vertices

If we knew the original \((x,y)\) coordinates of vertices, we would substitute them into the above - mentioned rules to get new coordinates.

Answer:

Without specific vertex coordinates in the figures, we can't provide exact new - vertex coordinates. But for a \(90^{\circ}\) counter - clockwise rotation about the origin, use \((x,y)\to(-y,x)\) and for a \(180^{\circ}\) rotation about the origin, use \((x,y)\to(-x,-y)\) to find new vertex coordinates.