Question

5.) rotate △npr 90° counterclockwise about the origin.
5b.) the coordinates of n’ are:
5c.) the length of (overline{pr}) is ______ units.
the length of (overline{pr}) is ______ units.
optional level 4: a.) rotate △klm 18

Explanation:

Step1: Recall rotation rule

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

Step2: Find coordinates of N'

Assume the coordinates of point N are $(x_N,y_N)$. After 90 - degree counter - clockwise rotation about the origin, the coordinates of N' are $(-y_N,x_N)$. From the graph, if we assume N has coordinates $(0, - 1)$, then N' has coordinates $(1,0)$.

Step3: Use distance formula for length of PR

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points P and R, if P has coordinates $(2, - 1)$ and R has coordinates $(2,-4)$, then $x_1 = 2,y_1=-1,x_2 = 2,y_2=-4$. Substitute into the formula: $d=\sqrt{(2 - 2)^2+(-4+1)^2}=\sqrt{0 + 9}=3$.

Step4: Property of rotation

A rotation is a rigid transformation, which preserves the lengths of line - segments. So the length of $\overline{P'R'}$ is the same as the length of $\overline{PR}$. So the length of $\overline{P'R'}$ is also 3 units.

Answer:

5b. $(1,0)$
5c. 3
5d. 3