Question

triangle abc has vertices a(1,2), b(3,4), and c(1,4). what are the coordinates of vertex a after the triangle is reflected across the y - axis, rotated 180 degrees about the origin, and then translated by (x,y)→(x + 3,y - 2)? a) (2,0) b) (4,0) c) (4,-4) d) (-2,0)

Explanation:

Step1: Reflect across y - axis

The rule for reflecting a point $(x,y)$ across the y - axis is $(x,y)\to(-x,y)$. For point $A(1,2)$, after reflection across the y - axis, it becomes $A_1(-1,2)$.

Step2: Rotate 180 degrees about origin

The rule for rotating a point $(x,y)$ 180 degrees about the origin is $(x,y)\to(-x,-y)$. For point $A_1(-1,2)$, after rotation, it becomes $A_2(1, - 2)$.

Step3: Translate

The translation rule is $(x,y)\to(x + 3,y-2)$. For point $A_2(1,-2)$, we have $x=1$ and $y = - 2$. Then $x+3=1 + 3=4$ and $y-2=-2-2=-4$. So the new point $A'$ is $(4,-4)$.

Answer:

C. $(4,-4)$