Question

problem solving6. what is the angle of rotation about the origin that maps $\triangle pqr$ to $\triangle p q r$?7. is $\triangle x y z$ a rotation of $\triangle xyz$? explain.8. $\triangle pqr$ is rotated $270^circ$ about the origin. graph and label the coordinates of $p$, $q$, and $r$.9. is $\triangle p q r$ a $270^circ$ rotation of $\triangle pqr$ about the origin? explain.

Explanation:

Problem 6

Step1: Identify original coordinates

Original $\triangle PQR$: $P(3,2)$, $Q(5,2)$, $R(5,6)$; Image $\triangle P'Q'R'$: $P'(-2,3)$, $Q'(-2,5)$, $R'(-6,5)$

Step2: Match rotation rule

Rule for 90° counterclockwise rotation: $(x,y)\to(-y,x)$
Test $P(3,2)$: $(-2,3)=P'$, $Q(5,2)$: $(-2,5)=Q'$, $R(5,6)$: $(-6,5)=R'$

A rotation preserves side lengths and the shape/orientation relationship of the figure. Compare $\triangle XYZ$ and $\triangle X'Y'Z'$: the side lengths of $\triangle XYZ$ are $\overline{XY}=2$, $\overline{XZ}=4$, $\overline{YZ}=\sqrt{2^2+4^2}=\sqrt{20}$; the side lengths of $\triangle X'Y'Z'$ are $\overline{X'Y'}=2$, $\overline{X'Z'}=4$, $\overline{Y'Z'}=\sqrt{4^2+2^2}=\sqrt{20}$, but the orientation does not match any rotation rule (90°, 180°, 270°) about the origin. $\triangle X'Y'Z'$ is a reflection and translation, not a rotation.

Step1: Identify original coordinates

Original $\triangle PQR$: $P(2,3)$, $Q(4,5)$, $R(2,6)$

Step2: Apply 270° rotation rule

Rule for 270° counterclockwise (or 90° clockwise) rotation about origin: $(x,y)\to(y,-x)$
$P(2,3)\to P'(3,-2)$
$Q(4,5)\to Q'(5,-4)$
$R(2,6)\to R'(6,-2)$

Step3: Plot new points

Plot $P'(3,-2)$, $Q'(5,-4)$, $R'(6,-2)$ on the grid and connect to form $\triangle P'Q'R'$

Answer:

$90^\circ$ counterclockwise (or $270^\circ$ clockwise)

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Problem 7