Question

follow the guided instructions below to rotate the figure 90° counter-clockwise about the origin. draw a circle centered at the center of rotation, such that one of the vertices of the figure is on the circle.

Explanation:

Step1: Identify Vertices

First, find the coordinates of the vertices of the original figure. Let's assume the vertices (from the graph) are: \( A(-7,1) \), \( B(-1,1) \), \( C(-1,8) \), \( D(-7,3) \) (we'll confirm the rotation rule).

Step2: Rotation Rule (90° Counter - Clockwise)

The rule for rotating a point \((x,y)\) 90° counter - clockwise about the origin is \((x,y)\to(-y,x)\).

For Point \( A(-7,1) \):

Apply the rule: \( x=-7,y = 1\). New coordinates: \((-1,-7)\)

For Point \( B(-1,1) \):

Apply the rule: \( x=-1,y = 1\). New coordinates: \((-1,-1)\)

For Point \( C(-1,8) \):

Apply the rule: \( x=-1,y = 8\). New coordinates: \((-8,-1)\)

For Point \( D(-7,3) \):

Apply the rule: \( x=-7,y = 3\). New coordinates: \((-3,-7)\)

Step3: Draw the Rotated Figure

Plot the new points \((-1,-7)\), \((-1,-1)\), \((-8,-1)\), \((-3,-7)\) on the coordinate plane and connect them to form the rotated figure. Also, draw a circle centered at the origin such that one of the vertices (e.g., the original or rotated) lies on the circle. For example, for the original point \( C(-1,8) \), the distance from the origin \((0,0)\) is \( \sqrt{(-1 - 0)^2+(8 - 0)^2}=\sqrt{1 + 64}=\sqrt{65}\). So the circle has radius \( \sqrt{65}\) and center at \((0,0)\).

(Note: Since the problem is about drawing, the final answer involves plotting the rotated points and the circle. If we consider the key part of the rotation calculation for the vertices, the rotated vertices are as calculated above.)

Answer:

The rotated figure's vertices (after 90° counter - clockwise rotation about the origin) are \((-1,-7)\), \((-1,-1)\), \((-8,-1)\), \((-3,-7)\) and the circle is centered at the origin with radius equal to the distance from the origin to any of the vertices (e.g., \(\sqrt{65}\) for vertex \( C(-1,8) \)). The figure is drawn by plotting these rotated points and the circle.