Question

3-3 additional practice
give the coordinates of each image. use a rule or graph paper to find your answer.

1. $r_{(90^{circ}, o)}(\triangle def)$ for $d(0, 5), e(-2, 8), f(-3, -5)$
2. $r_{(270^{circ}, o)}(wxyz)$ for $w(4, -2), x(7, 3), y(1, 11), z(-4, 6)$
3. $r_{(180^{circ}, o)}(\triangle stu)$ for $s(-2, -6), t(-5, 3), u(1, 0)$
4. $r_{(360^{circ}, o)}(jklm)$ for $j(-4, 7), k(1, 5), l(6, 1), m(3, -9)$

sat/act a point is rotated $270^{\circ}$ about the origin. the image of the point is $(-11, 7)$. what are the coordinates of the preimage?
\\(\boldsymbol{\text{ⓐ } (7, -11)}
\\(\boldsymbol{\text{ⓑ } (-7, -11)}
\\(\boldsymbol{\text{ⓒ } (7, 11)}
\\(\boldsymbol{\text{ⓓ } (11, 7)}
apply a blender has blades as shown. what rotation will map
the blade formed by $\triangle abc$ onto the blade formed by $\triangle def$?
image of a blender blade with triangles labeled a, b, c, d, e, f

Explanation:

Step1: Recall 90° rotation rule

For \( R_{(90^\circ, O)}(x,y) \), rule is \( (-y,x) \)

Step2: Apply rule to D(0,5)

\( (-5,0) \)

Step3: Apply rule to E(-2,8)

\( (-8,-2) \)

Step4: Apply rule to F(-3,-5)

\( (5,-3) \)

Step5: Recall 270° rotation rule

For \( R_{(270^\circ, O)}(x,y) \), rule is \( (y,-x) \)

Step6: Apply rule to W(4,-2)

\( (-2,-4) \)

Step7: Apply rule to X(7,3)

\( (3,-7) \)

Step8: Apply rule to Y(1,11)

\( (11,-1) \)

Step9: Apply rule to Z(-4,6)

\( (6,4) \)

Step10: Recall 180° rotation rule

For \( R_{(180^\circ, O)}(x,y) \), rule is \( (-x,-y) \)

Step11: Apply rule to S(-2,-6)

\( (2,6) \)

Step12: Apply rule to T(-5,3)

\( (5,-3) \)

Step13: Apply rule to U(1,0)

\( (-1,0) \)

Step14: Recall 360° rotation rule

For \( R_{(360^\circ, O)}(x,y) \), rule is \( (x,y) \) (no change)

Step15: Apply rule to J(-4,7)

\( (-4,7) \)

Step16: Apply rule to K(1,5)

\( (1,5) \)

Step17: Apply rule to L(6,1)

\( (6,1) \)

Step18: Apply rule to M(3,-9)

\( (3,-9) \)

Step19: Reverse 270° rotation for SAT/ACT

If \( R_{(270^\circ, O)}(a,b)=(-11,7) \), then \( (b,-a)=(-11,7) \). So \( b=-11 \), \( -a=7 \Rightarrow a=-7 \). Preimage is \( (-7,-11) \)

Answer:

1. D'(-5,0), E'(-8,-2), F'(5,-3)
2. W'(-2,-4), X'(3,-7), Y'(11,-1), Z'(6,4)
3. S'(2,6), T'(5,-3), U'(-1,0)
4. J'(-4,7), K'(1,5), L'(6,1), M'(3,-9)

SAT/ACT: B. (-7,-11)
(Note: The blender blade question lacks image details, so cannot be solved.)