The Endpoints Of The Diameter Of A Circle Are (-2, 3) And (-10, 9). Write An Equation Of This Circle In Standard Form And Identify Its Center And Radius.

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Introduction

In mathematics, a circle is a set of points that are equidistant from a central point called the center. The distance between any point on the circle and the center is called the radius. A diameter is a line segment that passes through the center of the circle and connects two points on the circle. The endpoints of the diameter are given as (-2, 3) and (-10, 9). In this article, we will find the equation of this circle in standard form and identify its center and radius.

Finding the Center of the Circle

To find the center of the circle, we need to find the midpoint of the diameter. The midpoint formula is given by:

(x1 + x2)/2, (y1 + y2)/2)

where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the diameter.

Let's substitute the given values into the midpoint formula:

(x1 + x2)/2 = (-2 + (-10))/2 = -12/2 = -6

(y1 + y2)/2 = (3 + 9)/2 = 12/2 = 6

Therefore, the center of the circle is at (-6, 6).

Finding the Radius of the Circle

The radius of the circle is the distance between the center and one of the endpoints of the diameter. We can use the distance formula to find the radius:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) and (x2, y2) are the coordinates of the center and one of the endpoints of the diameter.

Let's substitute the values into the distance formula:

d = √((-10 - (-6))^2 + (9 - 6)^2) d = √((-4)^2 + (3)^2) d = √(16 + 9) d = √25 d = 5

Therefore, the radius of the circle is 5.

Writing the Equation of the Circle in Standard Form

The standard form of the equation of a circle is given by:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) is the center of the circle and r is the radius.

Let's substitute the values into the standard form equation:

(x - (-6))^2 + (y - 6)^2 = 5^2 (x + 6)^2 + (y - 6)^2 = 25

Therefore, the equation of the circle in standard form is:

(x + 6)^2 + (y - 6)^2 = 25

Conclusion

In this article, we found the equation of a circle in standard form given the endpoints of the diameter. We also identified the center and radius of the circle. The center of the circle is at (-6, 6) and the radius is 5. The equation of the circle in standard form is:

(x + 6)^2 + (y - 6)^2 = 25

This equation can be used to graph the circle and find the coordinates of any point on the circle.

Applications of Circles in Real-World Scenarios

Circles have numerous applications in real-world scenarios, including:

  • Geometry and Trigonometry: Circles are used to study geometric shapes and trigonometric functions.
  • Physics and Engineering: Circles are used to model the motion of objects and design circular structures such as bridges and tunnels.
  • Computer Science: Circles are used in computer graphics and game development to create realistic images and animations.
  • Architecture: Circles are used in building design to create circular structures such as domes and arches.

Final Thoughts

In conclusion, the equation of a circle in standard form is a powerful tool for graphing and analyzing circular shapes. By understanding the properties of circles, we can apply them to real-world scenarios and solve complex problems. Whether you're a student, engineer, or architect, the study of circles is essential for understanding the world around us.

References

  • "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • "Trigonometry: A Unit Circle Approach" by Charles P. McKeague
  • "Physics for Scientists and Engineers" by Paul A. Tipler
  • "Computer Graphics: Principles and Practice" by James D. Foley

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources on the topic.

Introduction

Circles are a fundamental concept in mathematics, and understanding their properties and equations is essential for solving problems in various fields. In this article, we will address some frequently asked questions about circles, including their equations, properties, and applications.

Q: What is the equation of a circle in standard form?

A: The equation of a circle in standard form is given by:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) is the center of the circle and r is the radius.

Q: How do I find the center and radius of a circle given its equation?

A: To find the center and radius of a circle given its equation, you need to rewrite the equation in standard form. The center of the circle is the point (h, k) and the radius is the value of r.

Q: What is the difference between a circle and a sphere?

A: A circle is a two-dimensional shape that lies in a plane, while a sphere is a three-dimensional shape that is a set of points that are equidistant from a central point.

Q: How do I find the area of a circle?

A: The area of a circle is given by the formula:

A = πr^2

where r is the radius of the circle.

Q: How do I find the circumference of a circle?

A: The circumference of a circle is given by the formula:

C = 2Ï€r

where r is the radius of the circle.

Q: What is the relationship between the diameter and the radius of a circle?

A: The diameter of a circle is twice the radius, or:

d = 2r

Q: How do I find the equation of a circle given its diameter?

A: To find the equation of a circle given its diameter, you need to find the center of the circle and the radius. The center of the circle is the midpoint of the diameter, and the radius is half the length of the diameter.

Q: What are some real-world applications of circles?

A: Circles have numerous applications in real-world scenarios, including:

  • Geometry and Trigonometry: Circles are used to study geometric shapes and trigonometric functions.
  • Physics and Engineering: Circles are used to model the motion of objects and design circular structures such as bridges and tunnels.
  • Computer Science: Circles are used in computer graphics and game development to create realistic images and animations.
  • Architecture: Circles are used in building design to create circular structures such as domes and arches.

Q: How do I graph a circle?

A: To graph a circle, you need to plot the center of the circle and then draw a circle with the given radius.

Q: What is the equation of a circle with a center at (0, 0) and a radius of 5?

A: The equation of a circle with a center at (0, 0) and a radius of 5 is:

x^2 + y^2 = 25

Q: What is the equation of a circle with a center at (3, 4) and a radius of 2?

A: The equation of a circle with a center at (3, 4) and a radius of 2 is:

(x - 3)^2 + (y - 4)^2 = 4

Conclusion

In this article, we have addressed some frequently asked questions about circles, including their equations, properties, and applications. We hope that this article has provided you with a better understanding of circles and their importance in mathematics and real-world scenarios.

Final Thoughts

Circles are a fundamental concept in mathematics, and understanding their properties and equations is essential for solving problems in various fields. Whether you're a student, engineer, or architect, the study of circles is essential for understanding the world around us.

References

  • "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • "Trigonometry: A Unit Circle Approach" by Charles P. McKeague
  • "Physics for Scientists and Engineers" by Paul A. Tipler
  • "Computer Graphics: Principles and Practice" by James D. Foley

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources on the topic.