If The Scale Factor Between Two Circles Is 2 X 5 Y \frac{2x}{5y} 5 Y 2 X ​ , What Is The Ratio Of Their Areas?A. 2 X 5 Y \frac{2x}{5y} 5 Y 2 X ​ B. 2 X 2 5 Y 2 \frac{2x^2}{5y^2} 5 Y 2 2 X 2 ​ C. 4 X 2 Π 25 Y 2 \frac{4x^2\pi}{25y^2} 25 Y 2 4 X 2 Π ​ D. 4 X 2 25 Y 2 \frac{4x^2}{25y^2} 25 Y 2 4 X 2 ​

When dealing with geometric shapes, particularly circles, understanding the relationship between their scales and area ratios is crucial. In this article, we will delve into the concept of circle scales and explore how they relate to the area ratios of two circles. We will examine the given scale factor, $\frac{2x}{5y}$, and determine the correct ratio of their areas.

The Scale Factor and Its Significance

The scale factor between two circles is a measure of how much larger or smaller one circle is compared to the other. It is calculated by dividing the radius of the larger circle by the radius of the smaller circle. In this case, the scale factor is given as $\frac{2x}{5y}$. This means that the radius of the larger circle is $\frac{2x}{5y}$ times the radius of the smaller circle.

The Relationship Between Circle Scales and Area Ratios

The area of a circle is directly proportional to the square of its radius. This means that if the radius of a circle is multiplied by a factor, the area of the circle will be multiplied by the square of that factor. In the case of two circles with radii $r_1$ and $r_2$, the ratio of their areas is given by:

$$\frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2$$

where $A_1$ and $A_2$ are the areas of the two circles.

Applying the Scale Factor to the Area Ratio

Now that we have established the relationship between circle scales and area ratios, we can apply the given scale factor to determine the ratio of the areas of the two circles. The scale factor is $\frac{2x}{5y}$, which means that the radius of the larger circle is $\frac{2x}{5y}$ times the radius of the smaller circle.

Substituting this into the formula for the area ratio, we get:

$$\frac{A_1}{A_2} = \left(\frac{2x}{5y}\right)^2 = \frac{4x^2}{25y^2}$$

Conclusion

In conclusion, the ratio of the areas of two circles with a scale factor of $\frac{2x}{5y}$ is $\frac{4x^2}{25y^2}$. This means that if the radius of one circle is $\frac{2x}{5y}$ times the radius of the other circle, the area of the larger circle will be $\frac{4x^2}{25y^2}$ times the area of the smaller circle.

Answer

The correct answer is D. $\frac{4x^2}{25y^2}$.

Additional Information

• The scale factor between two circles is a measure of how much larger or smaller one circle is compared to the other.
• The area of a circle is directly proportional to the square of its radius.
• The ratio of the areas of two circles is given by the square of the scale factor.
• The scale factor can be used to determine the ratio of the areas of two circles.

References

• [1] "Geometry" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Related Topics

• Circle scales and area ratios
• Geometry and calculus
• Mathematics for computer science

Frequently Asked Questions

• Q: What is the scale factor between two circles? A: The scale factor between two circles is a measure of how much larger or smaller one circle is compared to the other.
• Q: How do you calculate the area ratio of two circles? A: The area ratio of two circles is given by the square of the scale factor.
• Q: What is the relationship between circle scales and area ratios? A: The area of a circle is directly proportional to the square of its radius.
  Circle Scales and Area Ratios: A Q&A Guide =============================================

In our previous article, we explored the relationship between circle scales and area ratios. We discussed how the scale factor between two circles can be used to determine the ratio of their areas. In this article, we will continue to delve into the world of circle scales and area ratios, answering some of the most frequently asked questions in this topic.

Q: What is the scale factor between two circles?

A: The scale factor between two circles is a measure of how much larger or smaller one circle is compared to the other. It is calculated by dividing the radius of the larger circle by the radius of the smaller circle.

Q: How do you calculate the area ratio of two circles?

A: The area ratio of two circles is given by the square of the scale factor. This means that if the scale factor between two circles is $\frac{2x}{5y}$, the area ratio of the two circles will be $\left(\frac{2x}{5y}\right)^2 = \frac{4x^2}{25y^2}$.

Q: What is the relationship between circle scales and area ratios?

A: The area of a circle is directly proportional to the square of its radius. This means that if the radius of a circle is multiplied by a factor, the area of the circle will be multiplied by the square of that factor.

Q: Can you give an example of how to use the scale factor to determine the area ratio of two circles?

A: Let's say we have two circles with radii 4 and 6. The scale factor between the two circles is $\frac{6}{4} = \frac{3}{2}$. To determine the area ratio of the two circles, we square the scale factor: $\left(\frac{3}{2}\right)^2 = \frac{9}{4}$. This means that the area of the larger circle is $\frac{9}{4}$ times the area of the smaller circle.

Q: What is the significance of the scale factor in real-world applications?

A: The scale factor has many real-world applications, particularly in engineering and architecture. For example, when designing a building, architects need to consider the scale factor between different components, such as the size of the foundation and the height of the building. By understanding the scale factor, architects can ensure that the building is structurally sound and aesthetically pleasing.

Q: Can you explain the concept of similar circles?

A: Similar circles are circles that have the same shape but different sizes. The scale factor between similar circles is the same for all corresponding points on the circles. This means that if we have two similar circles with radii 4 and 6, the scale factor between the two circles is $\frac{6}{4} = \frac{3}{2}$.

Q: How do you determine the scale factor between two similar circles?

A: To determine the scale factor between two similar circles, we divide the radius of the larger circle by the radius of the smaller circle. For example, if we have two similar circles with radii 4 and 6, the scale factor between the two circles is $\frac{6}{4} = \frac{3}{2}$.

Q: Can you give an example of how to use the scale factor to determine the area ratio of two similar circles?

A: Let's say we have two similar circles with radii 4 and 6. The scale factor between the two circles is $\frac{6}{4} = \frac{3}{2}$. To determine the area ratio of the two circles, we square the scale factor: $\left(\frac{3}{2}\right)^2 = \frac{9}{4}$. This means that the area of the larger circle is $\frac{9}{4}$ times the area of the smaller circle.

Conclusion

In conclusion, the scale factor between two circles is a measure of how much larger or smaller one circle is compared to the other. By understanding the scale factor, we can determine the area ratio of two circles and apply this knowledge to real-world applications. We hope that this Q&A guide has provided you with a better understanding of the concept of circle scales and area ratios.

Additional Resources

• [1] "Geometry" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Related Topics

• Circle scales and area ratios
• Geometry and calculus
• Mathematics for computer science

Frequently Asked Questions

• Q: What is the scale factor between two circles? A: The scale factor between two circles is a measure of how much larger or smaller one circle is compared to the other.
• Q: How do you calculate the area ratio of two circles? A: The area ratio of two circles is given by the square of the scale factor.
• Q: What is the relationship between circle scales and area ratios? A: The area of a circle is directly proportional to the square of its radius.