If You Use A Scale Factor Of $\frac{1}{5}$ To Create A Circle With A Radius Of 18 Cm, What Is The Radius Of The Original Circle You Dilated?A. 3.6 Cm B. 180 Cm C. 9 M D. 90 Cm
Introduction
In geometry, a scale factor is a ratio that describes the size change of a figure after a transformation. When dilating a circle, the scale factor is used to determine the new radius of the circle. In this article, we will explore how to use a scale factor to find the radius of the original circle after dilation.
What is a Scale Factor?
A scale factor is a ratio that describes the size change of a figure after a transformation. It is calculated by dividing the length of the new figure by the length of the original figure. In the case of a circle dilation, the scale factor is used to determine the new radius of the circle.
Understanding Circle Dilations
A circle dilation is a transformation that changes the size of a circle. When a circle is dilated, the center of the circle remains the same, but the radius of the circle changes. The scale factor is used to determine the new radius of the circle.
Calculating the Radius of the Original Circle
To calculate the radius of the original circle, we need to use the scale factor and the new radius of the circle. The formula for calculating the radius of the original circle is:
R = (r / s)
Where:
- R is the radius of the original circle
- r is the new radius of the circle
- s is the scale factor
Applying the Formula
In this problem, we are given a scale factor of 1/5 and a new radius of 18 cm. We need to calculate the radius of the original circle.
R = (18 / (1/5)) R = 18 x 5 R = 90 cm
Conclusion
In conclusion, we have learned how to use a scale factor to calculate the radius of the original circle after dilation. By applying the formula R = (r / s), we can determine the radius of the original circle. In this problem, we found that the radius of the original circle is 90 cm.
Answer
The correct answer is D. 90 cm.
Additional Examples
Here are a few additional examples of how to use a scale factor to calculate the radius of the original circle:
- If a circle is dilated with a scale factor of 2/3 and a new radius of 12 cm, what is the radius of the original circle?
- If a circle is dilated with a scale factor of 3/4 and a new radius of 20 cm, what is the radius of the original circle?
- If a circle is dilated with a scale factor of 1/2 and a new radius of 15 cm, what is the radius of the original circle?
Solutions
- R = (12 / (2/3)) R = 12 x (3/2) R = 18 cm
- R = (20 / (3/4)) R = 20 x (4/3) R = 26.67 cm
- R = (15 / (1/2)) R = 15 x 2 R = 30 cm
Conclusion
Q: What is a scale factor in geometry?
A: A scale factor is a ratio that describes the size change of a figure after a transformation. It is calculated by dividing the length of the new figure by the length of the original figure.
Q: How do you calculate the scale factor?
A: To calculate the scale factor, you need to divide the length of the new figure by the length of the original figure. For example, if the new radius of a circle is 18 cm and the original radius is 9 cm, the scale factor would be 18/9 = 2.
Q: What is the formula for calculating the radius of the original circle?
A: The formula for calculating the radius of the original circle is:
R = (r / s)
Where:
- R is the radius of the original circle
- r is the new radius of the circle
- s is the scale factor
Q: How do you apply the formula to find the radius of the original circle?
A: To apply the formula, you need to substitute the values of r and s into the equation. For example, if the new radius of a circle is 18 cm and the scale factor is 1/5, the radius of the original circle would be:
R = (18 / (1/5)) R = 18 x 5 R = 90 cm
Q: What if the scale factor is a fraction?
A: If the scale factor is a fraction, you need to invert the fraction and multiply it by the new radius. For example, if the scale factor is 2/3 and the new radius is 12 cm, the radius of the original circle would be:
R = (12 / (2/3)) R = 12 x (3/2) R = 18 cm
Q: Can you give me an example of a circle dilation with a scale factor of 3/4?
A: Yes, here's an example:
Suppose a circle is dilated with a scale factor of 3/4 and a new radius of 20 cm. To find the radius of the original circle, you would use the formula:
R = (r / s) R = (20 / (3/4)) R = 20 x (4/3) R = 26.67 cm
Q: What if the scale factor is a decimal?
A: If the scale factor is a decimal, you can simply divide the new radius by the scale factor. For example, if the scale factor is 0.5 and the new radius is 15 cm, the radius of the original circle would be:
R = (15 / 0.5) R = 30 cm
Q: Can you give me a summary of the key concepts?
A: Yes, here's a summary:
- A scale factor is a ratio that describes the size change of a figure after a transformation.
- The formula for calculating the radius of the original circle is R = (r / s).
- To apply the formula, you need to substitute the values of r and s into the equation.
- If the scale factor is a fraction, you need to invert the fraction and multiply it by the new radius.
- If the scale factor is a decimal, you can simply divide the new radius by the scale factor.
Conclusion
In conclusion, we have covered the key concepts of scale factors and circle dilations. We have learned how to calculate the scale factor, apply the formula to find the radius of the original circle, and handle different types of scale factors. We hope this article has been helpful in understanding these important concepts in geometry.