Question

a triangle with vertices at g(1, - 3), h(4, - 3), and i(1, - 1) is reflected across the line x = 2, then translated by the rule (x,y)→(x - 3,y + 5), and finally rotated 180 counter - clockwise about the origin. what are the coordinates of the final image of vertex h?

• a) h(3, - 2)
• b) h(3,2)
• c) h(-3,2)
• d) h(-3, - 2)

a) h(3, - 2)
b) h(3,2)
c) h(-3,2)
d) h(-3, - 2)

Explanation:

Step1: Reflect point H(4, - 3) across x = 2

The distance between x = 4 and x = 2 is $4 - 2=2$. Reflecting across x = 2, the new x - coordinate is $2-(4 - 2)=0$, and the y - coordinate remains the same. So the new point H' is (0, - 3).

Step2: Translate point H'

Using the translation rule $(x,y)\to(x - 3,y + 5)$, for point H'(0, - 3), the new x - coordinate is $0-3=-3$, and the new y - coordinate is $-3 + 5 = 2$. So the new point H'' is (-3, 2).

Step3: Rotate point H'' 180° counter - clockwise about the origin

The rule for a 180° counter - clockwise rotation about the origin is $(x,y)\to(-x,-y)$. For point H''(-3, 2), the new x - coordinate is $-(-3)=3$, and the new y - coordinate is $-2$. So the final point H''' is (3, - 2).

Answer:

A. H'''(3, - 2)