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 - degree counter - clockwise rotation about a point $(a,b)$ has the transformation rule $(x,y)\to(y - b+a,-x + a + b)$.

Step2: Apply rule to each vertex

Let the vertices of the figure be $(x_i,y_i)$. For each vertex, use the formula $(x_i,y_i)\to(y_i+ 1-2,-x_i-2 - 1)=(y_i - 1,-x_i - 3)$. Calculate the new coordinates of all vertices.

Step3: Plot new figure

Plot the new vertices obtained in the previous step and connect them to form the rotated figure. Also, to draw the circle as instructed, measure the distance from the center of rotation $(-2,-1)$ to one of the vertices of the original figure using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, where $(x_1,y_1)=(-2,-1)$ and $(x_2,y_2)$ is a vertex of the original figure. Then, with the center at $(-2,-1)$ and radius equal to $d$, draw the circle.

Answer:

The new figure with vertices obtained from the rotation rule and the circle centered at $(-2,-1)$ with appropriate radius as described above.