Answer :
Certainly! Let's determine the ratio of the areas of two circles given that the scale factor between their radii is \(\frac{2x}{5y}\).
1. Identify the scale factor:
- The given scale factor between the radii of the two circles is \(\frac{2x}{5y}\).
2. Understand the relationship between the scale factor and areas:
- The ratio of the areas of two similar shapes (such as circles) is the square of the ratio of their corresponding linear dimensions (in this case, their radii).
3. Calculate the ratio of the areas:
- The scale factor between the radii is \(\frac{2x}{5y}\). To find the ratio of their areas, we need to square this scale factor.
4. Square the scale factor:
[tex]\[ \left( \frac{2x}{5y} \right)^2 = \frac{(2x)^2}{(5y)^2} \][/tex]
5. Simplify the squared terms:
[tex]\[ \frac{(2x)^2}{(5y)^2} = \frac{4x^2}{25y^2} \][/tex]
Therefore, the ratio of the areas of the two circles is \(\frac{4x^2}{25y^2}\).
Among the given options, the correct answer is:
- \(\frac{4 x^2}{25 y^2}\)
So, the ratio of the areas of the two circles is [tex]\(\frac{4x^2}{25y^2}\)[/tex].
1. Identify the scale factor:
- The given scale factor between the radii of the two circles is \(\frac{2x}{5y}\).
2. Understand the relationship between the scale factor and areas:
- The ratio of the areas of two similar shapes (such as circles) is the square of the ratio of their corresponding linear dimensions (in this case, their radii).
3. Calculate the ratio of the areas:
- The scale factor between the radii is \(\frac{2x}{5y}\). To find the ratio of their areas, we need to square this scale factor.
4. Square the scale factor:
[tex]\[ \left( \frac{2x}{5y} \right)^2 = \frac{(2x)^2}{(5y)^2} \][/tex]
5. Simplify the squared terms:
[tex]\[ \frac{(2x)^2}{(5y)^2} = \frac{4x^2}{25y^2} \][/tex]
Therefore, the ratio of the areas of the two circles is \(\frac{4x^2}{25y^2}\).
Among the given options, the correct answer is:
- \(\frac{4 x^2}{25 y^2}\)
So, the ratio of the areas of the two circles is [tex]\(\frac{4x^2}{25y^2}\)[/tex].
