Question

1. triangle def has vertices d(-4,1), e(-2,1), and f(-3,3). the triangle is reflected across the x - axis, rotated 90 degrees counterclockwise about the origin, and then translated by (x,y)→(x - 1,y + 2). what are the coordinates of vertex e?

a) (0,-2)
b) (2,0)
c) (0,-4)
d) (0,0)
a) (0, 2)
b) (2,0)
c) (0,-4)
d) (0,0)

Explanation:

Step1: Reflection across x - axis

The rule for reflecting a point $(x,y)$ across the $x$-axis is $(x,-y)$. For point $E(-2,1)$, after reflection across the $x$-axis, the new point $E_1$ is $(-2, - 1)$.

Step2: Rotation 90 degrees counter - clockwise about the origin

The rule for rotating a point $(x,y)$ 90 degrees counter - clockwise about the origin is $(-y,x)$. For point $E_1(-2,-1)$, after rotation, the new point $E_2$ is $(1,-2)$.

Step3: Translation

The translation rule is $(x,y)\to(x - 1,y + 2)$. For point $E_2(1,-2)$, we have $x=1$ and $y = - 2$. After translation, $x'=1-1=0$ and $y'=-2 + 2=0$. So the coordinates of $E'$ are $(0,0)$.

Answer:

D. $(0,0)$