Question

1. quadrilateral efgh; 180°

e(4,1), f(3,-1), g(-2,3), h(-4,1)

1. quadrilateral pqrs; 270°
2. △abc; 270°
3. △rst; 90°

draw the image of the figure under the given rotation.

1. △pqr; 90°
2. quadrilateral defg; 270°

Explanation:

Step1: Recall rotation rules (origin center)

Rotation rules about origin:

• $180^\circ$: $(x,y) \to (-x,-y)$
• $90^\circ$ clockwise: $(x,y) \to (y,-x)$; $90^\circ$ counterclockwise: $(x,y) \to (-y,x)$
• $270^\circ$ clockwise: $(x,y) \to (-y,x)$; $270^\circ$ counterclockwise: $(x,y) \to (y,-x)$

Step2: Apply to Q3 (EFGH, $180^\circ$)

Given $E(4,1), F(3,-1), G(-2,3), H(-4,1)$:
$E'(-4,-1), F'(-3,1), G'(2,-3), H'(4,-1)$

Step3: Apply to Q4 (PQRS, $270^\circ$ clockwise)

Identify $P(-4,-3), Q(-1,-2), R(2,-3), S(-3,-4)$:
$P'(3,-4), Q'(2,-1), R'(3,2), S'(4,-3)$

Step4: Apply to Q5 ($\triangle ABC$, $270^\circ$ counterclockwise)

Identify $A(-1,3), B(-1,0), C(-3,2)$:
$A'(3,1), B'(0,1), C'(2,3)$

Step5: Apply to Q6 ($\triangle RST$, $90^\circ$ clockwise)

Identify $R(3,0), S(2,-4), T(4,-4)$:
$R'(0,-3), S'(-4,-2), T'(-4,-4)$

Step6: Apply to Q7 ($\triangle PQR$, $90^\circ$ counterclockwise)

Identify $P(1,1), Q(3,1), R(3,-4)$:
$P'(-1,1), Q'(-1,3), R'(4,3)$

Step7: Apply to Q8 (DEFG, $270^\circ$ clockwise)

Identify $D(-3,1), E(-1,0), F(-1,-3), G(-3,-2)$:
$D'(-1,-3), E'(0,-1), F'(3,-1), G'(2,-3)$

Answer:

1. Quadrilateral EFGH (180° rotation):

Image vertices: $E'(-4,-1)$, $F'(-3,1)$, $G'(2,-3)$, $H'(4,-1)$; connect to form the rotated quadrilateral.

1. Quadrilateral PQRS (270° clockwise rotation):

Image vertices: $P'(3,-4)$, $Q'(2,-1)$, $R'(3,2)$, $S'(4,-3)$; connect to form the rotated quadrilateral.

1. $\triangle ABC$ (270° counterclockwise rotation):

Image vertices: $A'(3,1)$, $B'(0,1)$, $C'(2,3)$; connect to form the rotated triangle.

1. $\triangle RST$ (90° clockwise rotation):

Image vertices: $R'(0,-3)$, $S'(-4,-2)$, $T'(-4,-4)$; connect to form the rotated triangle.

1. $\triangle PQR$ (90° counterclockwise rotation):

Image vertices: $P'(-1,1)$, $Q'(-1,3)$, $R'(4,3)$; connect to form the rotated triangle.

1. Quadrilateral DEFG (270° clockwise rotation):

Image vertices: $D'(-1,-3)$, $E'(0,-1)$, $F'(3,-1)$, $G'(2,-3)$; connect to form the rotated quadrilateral.

To plot these, mark each new vertex on the corresponding grid and draw the sides connecting the vertices in order.