Question

1. consider rectangle klmn.

a. rotate rectangle klmn 90 degrees counterclockwise about the origin.
b. what do you notice about the ordered pairs of the original figure and the ordered pairs of the rotation about the origin?

Explanation:

Step1: Recall rotation rule for 90 - degree counter - clockwise

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

Step2: Identify vertices of rectangle KLMN

Let's assume the coordinates of the vertices of rectangle KLMN are $K(x_1,y_1)$, $L(x_2,y_2)$, $M(x_3,y_3)$, $N(x_4,y_4)$. After applying the 90 - degree counter - clockwise rotation rule, the new vertices will be $K'(-y_1,x_1)$, $L'(-y_2,x_2)$, $M'(-y_3,x_3)$, $N'(-y_4,x_4)$.

Step3: Observe ordered - pair relationship

When we compare the original ordered pairs $(x,y)$ of the vertices of the rectangle and the rotated ordered pairs $(-y,x)$, we notice that the $x$ and $y$ values are transformed according to the rotation rule. The $x$ - coordinate of the original point becomes the $y$ - coordinate of the rotated point and the $y$ - coordinate of the original point changes sign and becomes the $x$ - coordinate of the rotated point.

Answer:

a. After rotating rectangle KLMN 90 degrees counter - clockwise about the origin, if the original coordinates of a vertex are $(x,y)$, the new coordinates are $(-y,x)$.
b. The $x$ and $y$ values of the ordered pairs of the vertices change according to the rule $(x,y)\to(-y,x)$. The sign of the $y$ - value of the original point changes and its position switches with the $x$ - value to form the new ordered pair for the rotated point.