If the scale factor between two circles is [tex]$\frac{2 x}{5 y}$[/tex], what is the ratio of their areas?
[tex]$\frac{2 x}{5 y}$[/tex]
[tex]$\frac{2 x^2}{5 y^2}$[/tex]
[tex]$\frac{4 x^2 \pi}{25 y^2}$[/tex]
[tex]$\frac{4 x^2}{25 y^2}$[/tex]
Answer (1)
The scale factor between two circles is 5 y 2 x .
The ratio of the areas of two similar figures is the square of the scale factor.
Square the scale factor: ( 5 y 2 x ) 2 = 25 y 2 4 x 2 .
The ratio of the areas is 25 y 2 4 x 2 .
Explanation
Problem Analysis The problem states that the scale factor between two circles is 5 y 2 x . We need to find the ratio of their areas.
Finding the Ratio of Areas The ratio of the areas of two similar figures is the square of the scale factor between them. Therefore, to find the ratio of the areas of the two circles, we need to square the scale factor.
Calculation The scale factor is 5 y 2 x . Squaring this gives us: ( 5 y 2 x ) 2 = ( 5 y ) 2 ( 2 x ) 2 = 25 y 2 4 x 2 Thus, the ratio of the areas of the two circles is 25 y 2 4 x 2 .
Final Answer The ratio of the areas of the two circles is 25 y 2 4 x 2 .
Examples
Imagine you are designing two circular gardens. The scale factor between the radii of the two gardens is 5 y 2 x . This means that for every 5 y units of radius in the larger garden, the smaller garden has 2 x units of radius. The ratio of their areas, 25 y 2 4 x 2 , tells you how much more space you have in one garden compared to the other. For example, if x = 1 and y = 1 , the larger garden has approximately 6.25 times the area of the smaller garden, influencing how you plan your flower arrangements and pathways.
