Question

a triangle with vertices at d(-4,1), e(-1,1), and f(-1,4) is reflected across the x - axis, then rotated 270 counterclockwise about the origin, and finally translated by the rule (x,y)→(x + 6,y + 2). what are the coordinates of the final image of vertex d?
. a) d(1,-2)
. b) d(2,1)
. c) d(5,6)
. d) d(2,7)
o a) d(1,-2)
o b) d(2,1)
o c) d(5,6)
o d) d(2,7)

Explanation:

Step1: Reflect across x - axis

The rule for reflecting a point $(x,y)$ across the $x$-axis is $(x,-y)$.
For point $D(-4,1)$, after reflection across the $x$-axis, the new point $D_1$ is $(-4, - 1)$.

Step2: Rotate 270° counter - clockwise about origin

The rule for rotating a point $(x,y)$ 270° counter - clockwise about the origin is $(y,-x)$.
For point $D_1(-4,-1)$, after rotation, the new point $D_2$ is $(-1,4)$.

Step3: Translate using the rule $(x,y)\to(x + 6,y+2)$

For point $D_2(-1,4)$, we have $x=-1$ and $y = 4$.
Applying the translation rule: $x'=-1 + 6=5$ and $y'=4 + 2=6$.
The final point $D'''$ is $(5,6)$.

Answer:

C. $D'''(5,6)$