Question

follow the guided instructions below to rotate the figure 270° counter - clockwise about the point (-2, -1). draw a circle centered at the center of rotation, such that one of the vertices of the figure is on the circle.

Explanation:

Step1: Recall rotation rule

A 270 - counter - clockwise rotation about a point $(a,b)$ of a point $(x,y)$ is given by the transformation $(x,y)\to(y + b - a,-x + a + b)$.

Step2: Identify vertices of the figure

Let the vertices of the original polygon be $(x_1,y_1),(x_2,y_2),\cdots$. Here the center of rotation is $(a=-2,b = - 1)$.

Step3: Apply the rotation formula to each vertex

For a vertex $(x,y)$:
The new $x$ - coordinate is $y+( - 1)-( - 2)=y + 1$.
The new $y$ - coordinate is $-x+( - 2)+( - 1)=-x - 3$.
Calculate the new coordinates for all vertices of the given figure.

Step4: Plot the new figure

Connect the new vertices to form the rotated figure. Also, to draw the circle, use the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ to find the radius (distance from the center of rotation $(-2,-1)$ to one of the vertices of the original figure), and then use a compass to draw the circle centered at $(-2,-1)$ with that radius.

Answer:

The rotated figure and the circle as described above should be drawn on the coordinate - plane. The vertices of the rotated figure are found by applying the transformation $(x,y)\to(y + 1,-x - 3)$ to each vertex of the original figure.