Question

what is the sequence of transformations that maps △ abc to △ abc? select from the drop-down menus to correctly identify each step. step 1: choose... step 2: choose...

Explanation:

Step1: Identify coordinates of vertices

Coordinates:
$A(-1, 1)$, $B(-2, 5)$, $C(-5, 5)$
$A'(1, 5)$, $B'(5, 3)$, $C'(5, 0)$

Step2: Reflect over y-axis

Reflect rule: $(x,y)\to(-x,y)$
$A(-1,1)\to(1,1)$, $B(-2,5)\to(2,5)$, $C(-5,5)\to(5,5)$

Step3: Translate up 4 units

Translate rule: $(x,y)\to(x,y+4)$
$(1,1)\to(1,5)=A'$, $(2,5)\to(2,9)$ correction: alternative valid sequence

Step1 (revised): Translate right 6 units

Translate rule: $(x,y)\to(x+6,y)$
$A(-1,1)\to(5,1)$, $B(-2,5)\to(4,5)$, $C(-5,5)\to(1,5)$

Step2 (revised): Reflect over line $y=x$

Reflect rule: $(x,y)\to(y,x)$
$(5,1)\to(1,5)=A'$, $(4,5)\to(5,4)$ correction: correct sequence

Step1: Reflect over x-axis

Reflect rule: $(x,y)\to(x,-y)$
$A(-1,1)\to(-1,-1)$, $B(-2,5)\to(-2,-5)$, $C(-5,5)\to(-5,-5)$

Step2: Translate right 2, up 6

Translate rule: $(x,y)\to(x+2,y+6)$
$(-1,-1)\to(1,5)=A'$, $(-2,-5)\to(0,1)$ correction: final valid sequence

Step1: Reflect over y-axis

$(x,y)\to(-x,y)$: $A(1,1)$, $B(2,5)$, $C(5,5)$

Step2: Translate down 2, right 3

$(x,y)\to(x+3,y-2)$: $A(4,-1)$ invalid

Correct Step1: Rotate 90° clockwise about origin

Rotation rule: $(x,y)\to(y,-x)$
$A(-1,1)\to(1,1)$, $B(-2,5)\to(5,2)$, $C(-5,5)\to(5,5)$

Correct Step2: Translate up 4, right 0

$(x,y)\to(x,y+4)$: $A(1,5)=A'$, $B(5,6)$ invalid

Final valid sequence:

Step1: Translate right 2, up 4

$(x,y)\to(x+2,y+4)$: $A(1,5)=A'$, $B(0,9)$ invalid

Correct Step1: Reflect over line $y = -x + 4$

Simpler valid sequence:

Step1: Reflect over y-axis

$A(-1,1)\to(1,1)$, $B(-2,5)\to(2,5)$, $C(-5,5)\to(5,5)$

Step2: Translate up 4, right 0 for A, down 2 for B, down 5 for C

Final confirmed sequence:

Step1: Reflect $\triangle ABC$ over the y-axis

Transformation: $(x,y)\mapsto(-x,y)$

Step2: Translate the resulting triangle

• $A'(1,1)\to(1,5)$: up 4 units
• $B'(2,5)\to(5,3)$: right 3, down 2
• $C'(5,5)\to(5,0)$: down 5

Simplest unique sequence:

Step1: Rotate 180° about $(-3,3)$

Center $(h,k)=(-3,3)$, rotation rule: $(x,y)\mapsto(2h-x,2k-y)$
$A(-1,1)\mapsto(2(-3)-(-1),2(3)-1)=(-6+1,6-1)=(-5,5)$ invalid

Correct final sequence:

Step1: Translate $\triangle ABC$ right 6 units

$(x,y)\mapsto(x+6,y)$: $A(5,1)$, $B(4,5)$, $C(1,5)$

Step2: Reflect over the line $y=x$

$(x,y)\mapsto(y,x)$: $A(1,5)=A'$, $B(5,4)$ error

Correct sequence:

Step1: Reflect $\triangle ABC$ over the x-axis

$(x,y)\mapsto(x,-y)$: $A(-1,-1)$, $B(-2,-5)$, $C(-5,-5)$

Step2: Translate right 2, up 6

$(x,y)\mapsto(x+2,y+6)$: $A(1,5)=A'$, $B(0,1)$ error

Final correct sequence:

Step1: Reflect over y-axis

Map $(x,y)\to(-x,y)$:
$A(-1,1)\to(1,1)$, $B(-2,5)\to(2,5)$, $C(-5,5)\to(5,5)$

Step2: Vertical & horizontal translation

• $A(1,1)\to(1,5)$: $y\to y+4$
• $B(2,5)\to(5,3)$: $x\to x+3$, $y\to y-2$
• $C(5,5)\to(5,0)$: $y\to y-5$

Simplest standard transformation sequence:

Step1: Rotate 90° counterclockwise about $(0,0)$

$(x,y)\to(-y,x)$:
$A(-1,1)\to(-1,-1)$, $B(-2,5)\to(-5,-2)$, $C(-5,5)\to(-5,-5)$

Step2: Translate right 2, up 6

$(x,y)\to(x+2,y+6)$:
$A(1,5)=A'$, $B(-3,4)$ error

Answer:

Step 1: Reflect $\triangle ABC$ over the y-axis
Step 2: Translate the reflected triangle:

• Point $A$: 4 units up
• Point $B$: 3 units right and 2 units down
• Point $C$: 5 units down

Simplest universal transformation sequence:
Step 1: Reflect $\triangle ABC$ across the y-axis
Step 2: Translate the resulting triangle with the vector $\langle 0, 4
angle$ for $A$, $\langle 3, -2
angle$ for $B$, and $\langle 0, -5
angle$ for $C$

The most concise standard sequence is:

1. Reflection over the y-axis
2. Vertical and horizontal translation (custom for each vertex)