How to Determine the Diameter of a Circle Given End and Center Points

When trying to determine the diameter of a circle given specific points, whether they are coordinates or other defined points, there are several methods that can be employed. In this article, we will explore how to calculate the diameter of a circle when given the end point with a value of 82 and the center point with a value of 64. We will discuss three different approaches: direct calculation using coordinate geometry, the Pythagorean theorem, and a step-by-step derivation to find the required diameter.

Direct Calculation Using Coordinate Geometry

The simplest method to find the diameter of a circle when given the end and center points is through the use of coordinate geometry. The diameter can be found by first determining the radius, which is the distance between the center point and the endpoint. Once the radius is known, the diameter can be calculated by multiplying it by 2.

Given the center point with coordinates (64, 62) and the end point with coordinates (82, 42), the first step is to find the distance between these two points. This distance is the magnitude of the radius.

Calculation of Radius:

The formula for the distance (radius) between two points (x1, y1) and (x2, y2) is:

R √[(82 - 64)2 (42 - 62)2]R sqrt{(82 - 64)^2 (42 - 62)^2}

R √[182 (-20)2]R sqrt{18^2 (-20)^2}

R √[324 400]R sqrt{324 400}

R √724R sqrt{724}

R 2√181R 2sqrt{181}

The diameter is then calculated by multiplying the radius by 2:

D 2 × 2√181D 2 times 2sqrt{181}

D 4√181D 4sqrt{181}

Using the Pythagorean Theorem

Another method to determine the diameter is by using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Here, the radius can be considered as one of the legs of the right-angled triangle, and the distance between the x-coordinates and the distance between the y-coordinates as the other two sides.

Calculation Using Pythagorean Theorem:

The distance between the x-coordinates (64 and 82) is 18, and the distance between the y-coordinates (62 and 42) is 20. Using the Pythagorean theorem:

R √182 202R sqrt{18^2 20^2}

R √(324 400)R sqrt{324 400}

R √724R sqrt{724}

R 2√181R 2sqrt{181}

The diameter is then:

D 2 × 2√181D 2 times 2sqrt{181}

D 4√181D 4sqrt{181}

Step-by-Step Derivation

Let's consider the step-by-step derivation using the Pythagorean theorem for clarity:

Calculate the distance between the x-coordinates: x 82 - 64 18x 82 - 64 18 Calculate the distance between the y-coordinates: y 62 - 42 20y 62 - 42 20 Apply the Pythagorean theorem: R √(x2 y2)R sqrt{x^2 y^2} R √(182 202)R sqrt{18^2 20^2} R √724R sqrt{724} R 2√181R 2sqrt{181}

Calculate the diameter:

D 2 × RD 2 times R

D 2 × 2√181D 2 times 2sqrt{181}

D 4√181D 4sqrt{181}

Conclusion

In this article, we have explored three methods to determine the diameter of a circle given the end and center points. Whether using coordinate geometry, the Pythagorean theorem, or a step-by-step derivation, the final result remains the same. The diameter of the circle is 4√181. The use of these methods not only helps in solving the problem but also reinforces the understanding of the underlying mathematical principles.