Question

a sequence of transformations is performed on △abc, resulting in △abc. each triangle is shown in the coordinate - plane below.
given transformations are listed below.

• $r_x$: a reflection across the x - axis
• $r_y$: a reflection across the y - axis
• $t_{-1,-7}$: a translation so that $(x,y)\to(x - 1,y - 7)$

which of these describes the sequence of transformations performed on △abc that results in △abc?
a. $t_{-1,-7}$ followed by $r_x$
b. $t_{-1,-7}$ followed by $r_y$
c. $r_x$ followed by $t_{-1,-7}$
d. $r_y$ followed by $t_{-1,-7}$

Explanation:

Step1: Analyze reflection properties

Reflection across the x - axis changes the sign of the y - coordinate (x,y)→(x, - y). Reflection across the y - axis changes the sign of the x - coordinate (x,y)→(-x,y). Translation T_{-1,-7} changes coordinates as (x,y)→(x - 1,y - 7).

Step2: Check option A

If we first apply T_{-1,-7} and then R_x. Let a point (x,y) be in △ABC. After T_{-1,-7}, it becomes (x - 1,y - 7). After R_x, it becomes (x - 1,-(y - 7))=(x - 1,-y + 7), which is not the correct transformation.

Step3: Check option B

If we first apply T_{-1,-7} to a point (x,y) in △ABC, we get (x - 1,y - 7). Then applying R_y gives (- (x - 1),y - 7)=(-x + 1,y - 7), which is not the correct transformation.

Step4: Check option C

If we first apply R_x to a point (x,y) in △ABC, we get (x,-y). Then applying T_{-1,-7} gives (x - 1,-y - 7), which is not the correct transformation.

Step5: Check option D

If we first apply R_y to a point (x,y) in △ABC, we get (-x,y). Then applying T_{-1,-7} gives (-x - 1,y - 7), which is consistent with the general pattern of transformation from △ABC to △A'B'C'.

Answer:

D. R_y followed by T_{-1,-7}