Question

Given that line segments are taken to line segments of the same length during rigid transformations, which transformation maps the line segment AB onto itself?

The line is -2,1 and 4,3
A) rotation counterclockwise of 90° → x-axis reflection → rotation counterclockwise of 270° → x-axis reflection
B) rotation counterclockwise of 90° → y-axis reflection → rotation counterclockwise of 270° → y-axis reflection
C) rotation counterclockwise of 90° → x-axis reflection → rotation counterclockwise of 270° → y-axis reflection
D) rotation counterclockwise of 180° → x-axis reflection → rotation counterclockwise of 270° → y-axis reflection

Answer 1

C) Rotation counter-clockwise of 90 degree → x-axis reflection → rotation counter-clockwise of 270 degree → y-axis reflection

Explanation:

We are given the line segment AB with end points (-2,1) and (4,3).

It is provided that after applying rigid transformations the line segment AB maps onto itself.

So, according to the options,

1. Rotation counter-clockwise by 90 degree changes (x, y) into (-y, x)

i.e. (-2,1) → (-1,-2) and (4,3) → (-3,4)

2. X-axis reflection changes (x, y) into (x, -y)

i.e. (-1,-2) → (-1,2) and (-3,4) → (-3,-4)

3. Rotation counter-clockwise by 270 degree changes (x,y) into (y, -x)

i.e. (-1,2) → (2,1) and (-3,-4) → (-4,3)

4. Y-axis reflection changes (x,y) into (-x,y)

i.e. (2,1) → (-2,1) and (-4,3) → (4,3)

Hence, we see that after applying transformations to AB we are again obtaining the line AB as seen below in the figure.

So, option C is correct.

Answer 2

Correct choice is C

Explanation:

Consider segment AB with endpoints A(-2,1) and B(4,3).

1. Rotation counterclockwise of 90° has a rule

(x,y)→(-y,x),

then

• A(-2,1)→C(-1,-2);
• B(4,3)→D(-3,4).

2. x-axis reflection has a rule

(x,y)→(x,-y),

then

• C(-1,-2)→E(-1,2);
• D(-3,4)→F(-3,-4).

3. Rotation counterclockwise of 270° has a rule

(x,y)→(y,-x),

then

• E(-1,2)→G(2,1);
• F(-3,-4)→H(-4,3).

4. y-axis reflection has a rule

(x,y)→(-x,y),

then

• G(2,1)→A(-2,1);
• H(-4,3)→B(4,3).