Question

9
a (0, 3) b (4, 2)
translation: (x,y) → (x - 3, y)
reflection: in x = 2
a ____ b ____
a ____ b ____
10
a (3, 5) b (2, 2)
translation: (x,y) → (x - 4, y)
rotation: 90° counterclockwise about the origin
a ____ b ____
a ____ b ____
describe the composition in words
11.
△abc → △abc ______
△abc → △abc ______
12.
△abc → △abc ______
△abc → △abc ______

Explanation:

Step1: Solve Problem 9 (Rotation first)

For point A(0, 5):

180° counterclockwise rotation rule: $(x,y) \to (-x,-y)$
$\boldsymbol{A'(0,5) \to (0,-5)}$

For point B(6, 2):

$\boldsymbol{B'(6,2) \to (-6,-2)}$

Step2: Problem9 (Reflection over $x=2$)

Reflection over $x=h$ rule: $(x,y) \to (2h-x,y)$ (here $h=2$)

For $A'(0,-5)$:

$\boldsymbol{A''=(2*2 - 0, -5)=(4,-5)}$

For $B'(-6,-2)$:

$\boldsymbol{B''=(2*2 - (-6), -2)=(10,-2)}$

Step3: Solve Problem10 (Translation first)

Translation rule: $(x,y) \to (x-4,y)$

For point A(3,5):

$\boldsymbol{A'=(3-4,5)=(-1,5)}$

For point B(7,2):

$\boldsymbol{B'=(7-4,2)=(3,2)}$

Step4: Problem10 (90° counterclockwise rotation)

90° counterclockwise rotation rule: $(x,y) \to (-y,x)$

For $A'(-1,5)$:

$\boldsymbol{A''=(-5,-1)}$

For $B'(3,2)$:

$\boldsymbol{B''=(-2,3)}$

Step5: Problem11 (Describe transformations)

$\triangle ABC \to \triangle A'B'C'$:

Rule for reflection over x-axis: $(x,y)\to(x,-y)$

$\triangle A'B'C' \to \triangle A''B''C''$:

Rule for horizontal translation right: $(x,y)\to(x+a,y)$ (right shift)

Step6: Problem12 (Describe transformations)

$\triangle ABC \to \triangle A'B'C'$:

Rule for 90° clockwise rotation about origin: $(x,y)\to(y,-x)$

$\triangle A'B'C' \to \triangle A''B''C''$:

Rule for horizontal translation right: $(x,y)\to(x+a,y)$ (right shift)

Answer:

Problem 9:

$A=(0,5)$, $B=(6,2)$
$A'=(0,-5)$, $B'=(-6,-2)$
$A''=(4,-5)$, $B''=(10,-2)$

Problem 10:

$A=(3,5)$, $B=(7,2)$
$A'=(-1,5)$, $B'=(3,2)$
$A''=(-5,-1)$, $B''=(-2,3)$

Problem 11:

$\triangle ABC \to \triangle A'B'C'$: Reflect $\triangle ABC$ across the x-axis.
$\triangle A'B'C' \to \triangle A''B''C''$: Translate $\triangle A'B'C'$ horizontally to the right.

Problem 12:

$\triangle ABC \to \triangle A'B'C'$: Rotate $\triangle ABC$ 90° clockwise about the origin.
$\triangle A'B'C' \to \triangle A''B''C''$: Translate $\triangle A'B'C'$ horizontally to the right.