Question

which set of transformations would bring triangle abc back onto itself exactly where it is.
rotate 180° about the origin, reflect over the y-axis, then translate 4 units to the right.
reflect over the x-axis, rotate 180° about the origin, translate 4 units to the right.
reflect over the x-axis, rotate 180° about the origin, reflect over the y-axis.
rotate 270° counter-clockwise, then reflect over the x-axis.

Explanation:

Step1: Analyze Triangle ABC Coordinates

First, identify coordinates of \( A(1,0) \), \( B(3,3) \), \( C(3,0) \).

Step2: Test Option 1

• Rotate \( 180^\circ \) about origin: \( (x,y) \to (-x,-y) \).

\( A(1,0) \to (-1,0) \), \( B(3,3) \to (-3,-3) \), \( C(3,0) \to (-3,0) \).

• Reflect over \( y \)-axis: \( (x,y) \to (-x,y) \).

\( (-1,0) \to (1,0) \), \( (-3,-3) \to (3,-3) \), \( (-3,0) \to (3,0) \).

• Translate 4 units right: \( (x,y) \to (x+4,y) \).

\( (1,0) \to (5,0) \), \( (3,-3) \to (7,-3) \), \( (3,0) \to (7,0) \).
Not original position. Eliminate.

Step3: Test Option 2

• Reflect over \( x \)-axis: \( (x,y) \to (x,-y) \).

\( A(1,0) \to (1,0) \), \( B(3,3) \to (3,-3) \), \( C(3,0) \to (3,0) \).

• Rotate \( 180^\circ \) about origin: \( (x,y) \to (-x,-y) \).

\( (1,0) \to (-1,0) \), \( (3,-3) \to (-3,3) \), \( (3,0) \to (-3,0) \).

• Translate 4 units right: \( (x,y) \to (x+4,y) \).

\( (-1,0) \to (3,0) \), \( (-3,3) \to (1,3) \), \( (-3,0) \to (1,0) \).
Not original position. Eliminate.

Step4: Test Option 3

• Reflect over \( x \)-axis: \( (x,y) \to (x,-y) \).

\( A(1,0) \to (1,0) \), \( B(3,3) \to (3,-3) \), \( C(3,0) \to (3,0) \).

• Rotate \( 180^\circ \) about origin: \( (x,y) \to (-x,-y) \).

\( (1,0) \to (-1,0) \), \( (3,-3) \to (-3,3) \), \( (3,0) \to (-3,0) \).

• Reflect over \( y \)-axis: \( (x,y) \to (-x,y) \).

\( (-1,0) \to (1,0) \), \( (-3,3) \to (3,3) \), \( (-3,0) \to (3,0) \).
Matches original coordinates \( A(1,0) \), \( B(3,3) \), \( C(3,0) \).

Step5: Test Option 4 (Optional)

• Rotate \( 270^\circ \) counter - clockwise: \( (x,y) \to (y,-x) \).

\( A(1,0) \to (0,-1) \), \( B(3,3) \to (3,-3) \), \( C(3,0) \to (0,-3) \).

• Reflect over \( x \)-axis: \( (x,y) \to (x,-y) \).

\( (0,-1) \to (0,1) \), \( (3,-3) \to (3,3) \), \( (0,-3) \to (0,3) \).
Not original position. Eliminate.

Answer:

C. Reflect over the x - axis, rotate \( 180^\circ \) about the origin, reflect over the y - axis.