Question

4 from unit 6, lesson 1
in the coordinate plane shown, reflect triangle $abc$ over the line $x = 0$. call this new triangle $abc$. then reflect triangle $abc$ over the line $y = 0$. call the resulting triangle $abc$.

which single transformation takes $abc$ to $abc$?
a translate triangle $abc$ by the directed line segment from $(1, 1)$ to $(-2, 1)$.
b reflect triangle $abc$ across the line $y = -x$.
c rotate triangle $abc$ counterclockwise using the origin as the center by $180^\circ$.
d dilate triangle $abc$ using the origin as the center and a scale factor of 2.

Explanation:

Step1: Identify coordinates of ABC

From the graph: $A(1,1)$, $B(2,-1)$, $C(3,0)$

Step2: Reflect over $x=0$ (y-axis)

Rule: $(x,y)\to(-x,y)$.
$A'(-1,1)$, $B'(-2,-1)$, $C'(-3,0)$

Step3: Reflect over $y=0$ (x-axis)

Rule: $(x,y)\to(x,-y)$.
$A''(-1,-1)$, $B''(-2,1)$, $C''(-3,0)$

Step4: Test each option

• Option A: Translation rule: $(x,y)\to(x-3,y)$. Result: $A(-2,1)$, $B(-1,-1)$, $C(0,0)$ (does not match $A''B''C''$)
• Option B: Reflect over $y=-x$: Rule $(x,y)\to(-y,-x)$. Result: $A(-1,-1)$, $B(1,-2)$, $C(0,-3)$ (does not match)
• Option C: Rotate 180° about origin: Rule $(x,y)\to(-x,-y)$. Result: $A(-1,-1)$, $B(-2,1)$, $C(-3,0)$ (matches $A''B''C''$)
• Option D: Dilate by scale factor 2: Rule $(x,y)\to(2x,2y)$. Result: $A(2,2)$, $B(4,-2)$, $C(6,0)$ (does not match)

Answer:

C. Rotate triangle ABC counterclockwise using the origin as the center by 180°.