Question

If a rectangle goes through a reflection, clockwise rotation of ${90}^o$, and then dilation with a scale factor of 5, then what will be the ratio of the area of the preimage to the area of image?

• A. 0.04
• B. 25
• C. 0.05
• D. 1/2

Answer 1

The correct option is A 0.04

It is given that the rectangle goes through a reflection, clockwise rotation of ${90}^o$ , and dilation with a scale factor of 5.

To calculate the ratio of the area of the preimage to the area of the image, we first need to calculate the area of the transformed rectangle and for that we need to calculate its length and width, as the area of a rectangle = length $\times$ width.


Let ABCD be the original rectangle and EFGH be the transformed image of ABCD.
Reflection and rotation will not produce any change in the areas, as these are the rigid transformations, only dilation will change the area of the image.

Now, area of ABCD = l w (Length of ABCD = l , × and width of ABCD = w)
And, area of EFGH = l’× w’ (Length of EFGH = l’, and width of EFGH = w’)

$\Rightarrow$ Ratio of the areas $=\frac{\text{Area of ABCD}}{\text{Area of EFGH}}=\frac{l\times w}{l^{\prime }\times w^{\prime }}..................\text{Equation 1}$

To get the ratio, we need to calculate the relation between $l,l^{\prime },w,$ and $w^{\prime }$.
(Side length of the image = Scale factor × Corresponding side length of the preimage)

$\Rightarrow EF=5\times AB$
Or $l^{\prime }=5\times l................\text{Equation 2}$
Similarly, $w^{\prime }=5\times w..............\text{Equation 3}$
On putting equations 2 and 3 in 1, we will get:
Ratio of area $=\frac{l\times w}{5\times l\times 5\times w}$
Or Ratio of area $=\frac{l\times w}{l^{\prime }\times w}=\frac{l\times w}{l\times w}\times \frac{1}{5\times 5}=\frac{1}{25}=0.04(\frac{l\times w}{l\times w}=1)$

Hence, the ratio of the area of the preimage to that of the image is 0.04.

➡Option D is correct.