Which composition of similarity transformations maps polygon ABCD to polygon A'B'C'D'?

a dilation with a scale factor of and then a rotation

a dilation with a scale factor of and then a translation

a dilation with a scale factor of 4 and then a rotation

a dilation with a scale factor of 4 and then a translation

Hey there!

Unless the smaller object was rotated 360° (in which case the rotation wouldn't have to be mentioned), you can see that all of the lines are still in the same place and that it wasn't rotated at all. This eliminated any answer option that mentions a rotation, which is A and C.

Also, if you count the units of one of the straight lines – for example, line AB and A'B' – you can see that the smaller object is four times smaller than the larger object. In the case of line AB and A'B', line AB is 8 units long and A'B' is 2 units long. This means that the scale factor is [tex] \frac{1}{4} [/tex].

Lastly, the smaller object was moved from its initial location, which would be in the center of the larger object if it wasn't moved after being scaled down.

The answer will be B, "a dilation with a scale factor of [tex] \frac{1}{4} [/tex] and then a translation."

Hope this helped you out! :-)

AkshayG AkshayG · Answer 2 · 2019-12-19T11:04:12+01:00

The composition of similarity transformations maps polygon ABCD to polygon A'B'C'D' is a dilation with a scale factor of [tex]\dfrac{1}{4}[/tex] and then a translation. Option (b) is correct.

Further explanation:

The rule of transformation of the coordinates can be expressed as follows,

[tex]\boxed{\left( {x,y} \right) \to \left( {kx,ky} \right)}[/tex]

Here, k represents the scale factor.

Given:

The options are as follows,

(a). A dilation with a scale factor of [tex]\dfrac{1}{4}[/tex] and then a rotation

(b). A dilation with a scale factor of [tex]\dfrac{1}{4}[/tex] and then a translation

(c). A dilation with a scale factor of 4 and then a rotation.

(d). A dilation with a scale factor of 4 and then a translation

Explanation:

The length of side [tex]{\text{A'B'}}[/tex] is [tex]\dfrac{1}{4}[/tex] of side AB.

The length of side [tex]{\text{A'D'}}[/tex] is [tex]\dfrac{1}{4}[/tex] of side AD.

The length of side [tex]{\text{D'B'}}[/tex] is [tex]\dfrac{1}{4}[/tex] of side DB.

The length of side [tex]{\text{C'B'}}[/tex] is [tex]\dfrac{1}{4}[/tex] of side CB.

The composition of similarity transformations maps polygon ABCD to polygon A'B'C'D' is a dilation with a scale factor of [tex]\dfrac{1}{4}[/tex] and then a translation. Option (b) is correct.

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Coordinate geometry

Keywords: composition, image vertices, dilation, center, scale factor, four, coordinates, x-coordinates, y-coordinate, transformation rule,