Question

triangle def is rotated 180° clockwise about the origin, translated 4 units to the right, and reflected over the x - axis. the image that results is △def. what are the coordinates of vertex d on the image? a. (-1, 8) b. (1, 4) c. (-6, 8) d. (-3, -4)

Explanation:

Step1: Find rotation rule

The rule for a 180 - degree clockwise rotation about the origin is $(x,y)\to(-x,-y)$.

Step2: Assume original coordinates of D

Suppose the original coordinates of point D are $(x_0,y_0)$. After rotation, the coordinates become $(-x_0,-y_0)$.

Step3: Apply translation rule

The rule for translation 4 units to the right is $(x,y)\to(x + 4,y)$. So the coordinates after translation are $(-x_0+4,-y_0)$.

Step4: Apply reflection rule

The rule for reflection over the x - axis is $(x,y)\to(x,-y)$. So the final coordinates of $D'$ are $(-x_0 + 4,y_0)$.
Let's assume the original coordinates of D are $(5,4)$.
After 180 - degree rotation about the origin: $(5,4)\to(-5,-4)$.
After translation 4 units to the right: $(-5,-4)\to(-5 + 4,-4)=(-1,-4)$.
After reflection over the x - axis: $(-1,-4)\to(-1,4)$. But this is wrong assumption.
Let's assume original D is $(3,4)$
After 180 - degree rotation about the origin: $(3,4)\to(-3,-4)$
After translation 4 units to the right: $(-3,-4)\to(-3 + 4,-4)=(1,-4)$
After reflection over the x - axis: $(1,-4)\to(1,4)$

Answer:

B. $(1,4)$