The Diameter Of A Circle Is 6 Cm. If One End Of The Diameter Is At { (-4,0)$}$, What Are The Coordinates Of The Other End On The { X$}$-axis?

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Introduction

In geometry, a circle is a set of points that are all equidistant from a central point called the center. The distance from the center to any point on the circle 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 length of the diameter is twice the length of the radius. In this article, we will explore how to find the coordinates of the other end of a diameter given one end and the diameter.

Understanding the Problem

We are given that the diameter of a circle is 6 cm and one end of the diameter is at (-4, 0). We need to find the coordinates of the other end of the diameter on the x-axis. To solve this problem, we can use the concept of the midpoint formula, which states that the coordinates of the midpoint of a line segment are the average of the coordinates of the two endpoints.

The Midpoint Formula

The midpoint formula is given by:

(x_m, y_m) = ((x_1 + x_2)/2, (y_1 + y_2)/2)

where (x_m, y_m) is the midpoint of the line segment, and (x_1, y_1) and (x_2, y_2) are the coordinates of the two endpoints.

Applying the Midpoint Formula

In this case, we know that the midpoint of the diameter is the center of the circle. Since the diameter is 6 cm, the radius is 3 cm. The center of the circle is 3 cm away from the point (-4, 0) in the x-direction. Therefore, the x-coordinate of the center is -4 + 3 = -1.

Finding the Coordinates of the Other End

Since the center of the circle is the midpoint of the diameter, we can use the midpoint formula to find the coordinates of the other end of the diameter. Let (x_2, y_2) be the coordinates of the other end of the diameter. Then, we have:

(-1, 0) = ((-4 + x_2)/2, (0 + y_2)/2)

Simplifying the equation, we get:

-1 = (-4 + x_2)/2 0 = (0 + y_2)/2

Solving for x_2, we get:

x_2 = 2(-1) + 4 x_2 = 2

Since the point (2, 0) lies on the x-axis, the coordinates of the other end of the diameter are (2, 0).

Conclusion

In this article, we used the concept of the midpoint formula to find the coordinates of the other end of a diameter given one end and the diameter. We applied the midpoint formula to find the coordinates of the center of the circle and then used the center to find the coordinates of the other end of the diameter. The coordinates of the other end of the diameter are (2, 0).

Frequently Asked Questions

  • What is the diameter of a circle? A: The diameter of a circle is a line segment that passes through the center of the circle and connects two points on the circle.
  • What is the radius of a circle? A: The radius of a circle is the distance from the center to any point on the circle.
  • How do you find the coordinates of the other end of a diameter? A: To find the coordinates of the other end of a diameter, you can use the midpoint formula to find the coordinates of the center of the circle and then use the center to find the coordinates of the other end of the diameter.

References

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

Further Reading

  • [1] "The Diameter of a Circle" by Math Open Reference
  • [2] "The Midpoint Formula" by Math Is Fun
  • [3] "Geometry and Trigonometry" by Khan Academy

Introduction

In our previous article, we explored how to find the coordinates of the other end of a diameter given one end and the diameter. We used the concept of the midpoint formula to find the coordinates of the center of the circle and then used the center to find the coordinates of the other end of the diameter. In this article, we will answer some frequently asked questions related to the diameter of a circle.

Q&A

Q: What is the diameter of a circle?

A: The diameter of a circle is a line segment that passes through the center of the circle and connects two points on the circle.

Q: What is the radius of a circle?

A: The radius of a circle is the distance from the center to any point on the circle.

Q: How do you find the coordinates of the other end of a diameter?

A: To find the coordinates of the other end of a diameter, you can use the midpoint formula to find the coordinates of the center of the circle and then use the center to find the coordinates of the other end of the diameter.

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

A: The diameter of a circle is twice the length of the radius.

Q: Can the diameter of a circle be negative?

A: No, the diameter of a circle cannot be negative. The diameter is a length, and lengths are always non-negative.

Q: Can the diameter of a circle be zero?

A: No, the diameter of a circle cannot be zero. A circle with a diameter of zero would not be a circle, but a point.

Q: How do you find the length of the diameter of a circle?

A: To find the length of the diameter of a circle, you can use the Pythagorean theorem. If you know the coordinates of two points on the circle, you can use the distance formula to find the length of the diameter.

Q: Can the diameter of a circle be a vector?

A: Yes, the diameter of a circle can be a vector. In this case, the diameter is a line segment that passes through the center of the circle and connects two points on the circle.

Q: Can the diameter of a circle be a function?

A: Yes, the diameter of a circle can be a function. In this case, the diameter is a function that takes a point on the circle as input and returns the length of the diameter.

Conclusion

In this article, we answered some frequently asked questions related to the diameter of a circle. We covered topics such as the definition of the diameter, the relationship between the diameter and the radius, and how to find the coordinates of the other end of a diameter.

Frequently Asked Questions

  • What is the diameter of a circle?
  • What is the radius of a circle?
  • How do you find the coordinates of the other end of a diameter?
  • What is the relationship between the diameter and the radius of a circle?
  • Can the diameter of a circle be negative?
  • Can the diameter of a circle be zero?
  • How do you find the length of the diameter of a circle?
  • Can the diameter of a circle be a vector?
  • Can the diameter of a circle be a function?

References

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

Further Reading

  • [1] "The Diameter of a Circle" by Math Open Reference
  • [2] "The Midpoint Formula" by Math Is Fun
  • [3] "Geometry and Trigonometry" by Khan Academy