Which sequences of transformations applied to shape i prove that shape i is similar to shape ii?

a. reflection across the x-axis, followed by a reflection across the y-axis, and then a dilation by a scale factor of 0.5
b. a 90° counterclockwise rotation about the origin, and then a dilation by a scale factor of 0.5
c. a 180° counterclockwise rotation about the origin, and then a dilation by a scale factor of 0.5
d. a translation 2 units down and 5 units left, and then a dilation by a scale factor of 0.5

The sequence of transformations applied to shape I is:

a. reflection across the x-axis, followed by a reflection across the y-axis, and then a dilation by a scale factor of 0.5

Step-by-step explanation:

Now when this point is reflected across x-axis the point is transformed to:

(2,-1)

( Since the rule of transformation is:

(x,y) → (x,-y) )

Now again when this point is reflected across the y-axis the point is transformed to:

(-2,-1)

( Since the rule of transformation is:

(x,y) → (-x,y) )

Now when this point is dilated by a scale factor of 0.5 the point that is obtained is:

(-1,-0.5)

( Since, the rule of transformation is:

(x,y) → (0.5x,0.5y) )

Hence, A (2,1) → A' (-1,-0.5)

Now when this point is reflected across x-axis the point is transformed to:

(3,-1)

( Since the rule of transformation is:

(x,y) → (x,-y) )

Now again when this point is reflected across the y-axis the point is transformed to:

(-3,-1)

( Since the rule of transformation is:

(x,y) → (-x,y) )

Now when this point is dilated by a scale factor of 0.5 the point that is obtained is:

(-1.5,-0.5)

( Since, the rule of transformation is:

(x,y) → (0.5x,0.5y) )

Hence, B (3,1) → B' (-1.5,-0.5)

Similarly all the other points of shape I could be mapped to obtain the shape II.

Hence, option: a is correct.

Option A and C

Step-by-step explanation:

We are given two shapes I and II.

A point in shape I is at (2,1).

The corresponding point in shape II is at (-1,-0.5).

We know that

The transformation rule when a point is reflected across x- axis is given by

By applying this rule on shape then we get

The transformation rule when a point is reflected across y- axis is given by

Apply this rule then we get

The transformation when a point is dilate by scale factor 0.5 is given by

Now, apply dilation by scale factor 0.5

Therefore, shape I transformed into shape I.

The transformation rule when a point is rotated 180 degree about origin is given by

By apply this rule on shape I

Then , the coordinates of corresponding point after applying the rule

Now, dilation applied on this point and the point is dilated by scale factor 0.5

Then , the coordinated of this point

Therefore,  shape I transformed into shape II.

Hence,option A and C is true.