Question

practice performing a rotation in the coordinate plane

study the example showing a 270° counterclockwise rotation around the origin. then solve problems 1 - 5.

example
samuel shows his step - dad a propeller design for a model plane. samuel explains how he rotated △lmn 270° counterclockwise around the origin to form △lmn. how might samuel have performed the rotation on △lmn to form △lmn?
samuels drawing shows a circle, centered at the origin and passing through vertex m of △lmn. since 270° is equal to three 90° turns, samuel could have rotated vertex m three 90° counterclockwise turns to vertex m.
based on the positions of m and l in △lmn, he could have counted 2 units up from m to find l. based on the positions of l and n in △lmn, he could have moved 1 unit up from l and then 1 unit to the left to find n.

1. compare the coordinates of the corresponding vertices of △lmn and △lmn in the example. describe the effect of the 270° counterclockwise rotation on the vertices. what other rotation of △lmn would have the same effect?
2. amal drew quadrilateral abcd in the coordinate plane. then, amal rotated abcd around the origin to draw quadrilateral abcd. draw a 180° rotation and a 90° clockwise rotation of quadrilateral abcd around the origin.

Explanation:

Step1: Recall rotation rules

For a 270 - counter - clockwise rotation about the origin, if a point $(x,y)$ is rotated, the new coordinates $(x',y')$ are given by $(y, - x)$.

Step2: Analyze effect on vertices

Let the coordinates of a vertex of $\triangle LMN$ be $(x,y)$. After a 270 - counter - clockwise rotation about the origin, its new coordinates in $\triangle L'M'N'$ will be $(y, - x)$. This means the $x$ and $y$ coordinates are swapped and the new $x$ - coordinate is negated.

Step3: Find equivalent rotation

A 90 - clockwise rotation about the origin also has the transformation rule $(x,y)\to(y, - x)$. So a 90 - clockwise rotation of $\triangle LMN$ would have the same effect as a 270 - counter - clockwise rotation.

Answer:

The effect of a 270 - counter - clockwise rotation on the vertices is that the $x$ and $y$ coordinates of each vertex are swapped and the new $x$ - coordinate is negated. A 90 - clockwise rotation of $\triangle LMN$ would have the same effect.