Ricardo Draws A Circle On The Coordinate Plane. The Circle Has The Equation $\sqrt{16 - (x+2)^2} - 5$. The Circle Is Now Translated 3 Units To The Right And 3 Units Up. Where Is The Translated Circle's Center?

Introduction

In mathematics, the translation of a geometric shape is a fundamental concept that involves moving the shape from one position to another without changing its size or orientation. In this article, we will explore the translation of a circle on the coordinate plane and determine the new center of the translated circle.

Understanding the Original Circle

The original circle has the equation $\sqrt{16 - (x+2)^2} - 5$. To understand this equation, let's break it down into its components. The expression inside the square root, $16 - (x+2)^2$, represents the squared distance between a point $(x, y)$ on the circle and the center of the circle. The $-5$ outside the square root represents the vertical distance between the center of the circle and the x-axis.

The Center of the Original Circle

To find the center of the original circle, we need to rewrite the equation in the standard form of a circle: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius. Let's start by isolating the squared term inside the square root:

$$\sqrt{16 - (x+2)^2} - 5 = 0$$

$$16 - (x+2)^2 = 25$$

$$(x+2)^2 = 9$$

$$x+2 = \pm 3$$

$$x = -2 \pm 3$$

$$x = 1 \text{ or } x = -5$$

Now, let's find the corresponding values of $y$:

$$y = \sqrt{16 - (x+2)^2} + 5$$

$$y = \sqrt{16 - (1+2)^2} + 5$$

$$y = \sqrt{16 - 9} + 5$$

$$y = \sqrt{7} + 5$$

$$y = 5 + \sqrt{7}$$

So, the center of the original circle is at the point $(-2, 5 + \sqrt{7})$.

The Translation of the Circle

The circle is translated 3 units to the right and 3 units up. This means that the new center of the circle will be 3 units to the right and 3 units up from the original center.

Finding the New Center

To find the new center of the circle, we need to add 3 to the x-coordinate and 3 to the y-coordinate of the original center:

$$x_{\text{new}} = x_{\text{original}} + 3$$

$$y_{\text{new}} = y_{\text{original}} + 3$$

$$x_{\text{new}} = -2 + 3$$

$$x_{\text{new}} = 1$$

$$y_{\text{new}} = 5 + \sqrt{7} + 3$$

$$y_{\text{new}} = 8 + \sqrt{7}$$

So, the new center of the circle is at the point $(1, 8 + \sqrt{7})$.

Conclusion

In this article, we have explored the translation of a circle on the coordinate plane and determined the new center of the translated circle. We have seen that the translation of a circle involves moving the circle from one position to another without changing its size or orientation. By understanding the original circle and its center, we can find the new center of the translated circle by adding the translation values to the original center.

Key Takeaways

• The translation of a circle involves moving the circle from one position to another without changing its size or orientation.
• To find the new center of a translated circle, we need to add the translation values to the original center.
• The new center of the circle is at the point $(1, 8 + \sqrt{7})$.

Further Exploration

• What happens when a circle is translated by a vector other than 3 units to the right and 3 units up?
• How does the translation of a circle affect its equation?
• Can we find the equation of the translated circle using the new center and the original radius?
  Ricardo's Circle Translation: Q&A =====================================

Introduction

In our previous article, we explored the translation of a circle on the coordinate plane and determined the new center of the translated circle. In this article, we will answer some frequently asked questions about the translation of a circle and provide additional insights into this important mathematical concept.

Q: What is the difference between a translation and a rotation?

A: A translation is a movement of a geometric shape from one position to another without changing its size or orientation. A rotation, on the other hand, is a movement of a geometric shape around a fixed point without changing its size or position.

Q: How do I find the new center of a translated circle?

A: To find the new center of a translated circle, you need to add the translation values to the original center. For example, if the original center is at the point $(x, y)$ and the translation is 3 units to the right and 3 units up, the new center will be at the point $(x + 3, y + 3)$.

Q: What happens when a circle is translated by a vector other than 3 units to the right and 3 units up?

A: When a circle is translated by a vector other than 3 units to the right and 3 units up, the new center of the circle will be different. For example, if the original center is at the point $(x, y)$ and the translation is 2 units to the right and 4 units up, the new center will be at the point $(x + 2, y + 4)$.

Q: How does the translation of a circle affect its equation?

A: The translation of a circle affects its equation by changing the center of the circle. The new equation of the circle will have the same radius as the original circle, but the center will be different.

Q: Can we find the equation of the translated circle using the new center and the original radius?

A: Yes, we can find the equation of the translated circle using the new center and the original radius. The equation of the translated circle will be in the form $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the new center and $r$ is the original radius.

Q: What is the relationship between the translation of a circle and the concept of symmetry?

A: The translation of a circle is related to the concept of symmetry. When a circle is translated, it remains symmetrical about its center. This means that if we reflect the circle about its center, we will get the same circle.

Q: Can we use the concept of translation to solve real-world problems?

A: Yes, we can use the concept of translation to solve real-world problems. For example, in architecture, we can use the concept of translation to design buildings and bridges that are symmetrical and aesthetically pleasing.

Conclusion

In this article, we have answered some frequently asked questions about the translation of a circle and provided additional insights into this important mathematical concept. We have seen that the translation of a circle involves moving the circle from one position to another without changing its size or orientation. By understanding the original circle and its center, we can find the new center of the translated circle and use this concept to solve real-world problems.

Key Takeaways

• The translation of a circle involves moving the circle from one position to another without changing its size or orientation.
• To find the new center of a translated circle, we need to add the translation values to the original center.
• The translation of a circle affects its equation by changing the center of the circle.
• We can find the equation of the translated circle using the new center and the original radius.
• The concept of translation is related to the concept of symmetry.