Question

draw and label the final image of $\triangle abc$ after the given sequence of transformations.

1. reflect $\triangle abc$ over the y-axis and then translate by $(2, -3)$.
2. rotate $\triangle abc$ 90 degrees clockwise about the origin and then reflect over the x-axis.
3. translate $\triangle abc$ by $(4, 4)$, rotate 90 degrees counterclockwise around $a$, and reflect over the y-axis.
4. reflect $\triangle abc$ over the x-axis, translate by $(-3, -1)$, and rotate 180 degrees around the origin.

online homework

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Explanation:

First, identify the coordinates of $\triangle ABC$ from the grid: $A(0,1)$, $B(-3,3)$, $C(-1,4)$

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Problem 1

Step1: Reflect over y-axis

Rule: $(x,y)\to(-x,y)$
$A(0,1)\to A'(0,1)$, $B(-3,3)\to B'(3,3)$, $C(-1,4)\to C'(1,4)$

Step2: Translate by $\langle2,-3

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

Step1: Rotate 90° clockwise about origin

Rule: $(x,y)\to(y,-x)$
$A(0,1)\to A'(1,0)$, $B(-3,3)\to B'(3,3)$, $C(-1,4)\to C'(4,1)$

Step2: Reflect over x-axis

Rule: $(x,y)\to(x,-y)$
$A'(1,0)\to A''(1,0)$
$B'(3,3)\to B''(3,-3)$
$C'(4,1)\to C''(4,-1)$

Step1: Translate by $\langle4,4

angle$
Rule: $(x,y)\to(x+4,y+4)$
$A(0,1)\to A'(0+4,1+4)=(4,5)$
$B(-3,3)\to B'(-3+4,3+4)=(1,7)$
$C(-1,4)\to C'(-1+4,4+4)=(3,8)$

Step2: Rotate 90° counterclockwise about $A'(4,5)$

First translate points so $A'$ is origin: $(x,y)\to(x-4,y-5)$
$A'(4,5)\to(0,0)$, $B'(1,7)\to(-3,2)$, $C'(3,8)\to(-1,3)$
Rotate 90° counterclockwise: $(x,y)\to(-y,x)$
$(0,0)\to(0,0)$, $(-3,2)\to(-2,-3)$, $(-1,3)\to(-3,-1)$
Translate back: $(x,y)\to(x+4,y+5)$
$A''(0+4,0+5)=(4,5)$
$B''(-2+4,-3+5)=(2,2)$
$C''(-3+4,-1+5)=(1,4)$

Step3: Reflect over y-axis

Rule: $(x,y)\to(-x,y)$
$A''(4,5)\to A'''(-4,5)$
$B''(2,2)\to B'''(-2,2)$
$C''(1,4)\to C'''(-1,4)$

Answer:

Final coordinates: $A''(2,-2)$, $B''(5,0)$, $C''(3,1)$ (plot these points to draw the final triangle)

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Problem 2