Introduction

Understanding how to find the center and radius of a circle when given the endpoints of its diameter is a fundamental concept in geometry. This article will walk you through a detailed example, employing the midpoint formula and the Pythagorean theorem to solve the problem. These methods are essential tools for anyone working on geometric problems, and they also enhance the optimization of content for search engines.

Given Endpoints of a Diameter

Suppose the endpoints of the diameter of a circle are (-7, 3) and (5, 1). Our goal is to determine the center of the circle and calculate its radius.

Step 1: Find the Center of the Circle

The center of the circle is the midpoint of the diameter. The midpoint formula for a segment with endpoints ( (x_1, y_1) ) and ( (x_2, y_2) ) is:

[ M left( frac{x_1 x_2}{2}, frac{y_1 y_2}{2} right) ]

Substituting the given points (-7, 3) and (5, 1) into the formula:

[ M left( frac{-7 5}{2}, frac{3 1}{2} right) ]

Calculating the x-coordinate:

[ frac{-7 5}{2} frac{-2}{2} -1 ]

Calculating the y-coordinate:

[ frac{3 1}{2} frac{4}{2} 2 ]

Therefore, the center of the circle is at the point ((-1, 2)).

Step 2: Find the Radius of the Circle

To find the radius, we first need to determine the length of the diameter using the distance formula, which is:

[ d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2} ]

Substituting the given points (-7, 3) and (5, 1) into the formula:

[ d sqrt{(5 - (-7))^2 (1 - 3)^2} ]

Calculating the x-coordinate difference:

[ 5 - (-7) 5 7 12 ]

Calculating the y-coordinate difference:

[ 1 - 3 -2 ]

Substituting these differences into the distance formula:

[ d sqrt{12^2 (-2)^2} sqrt{144 4} sqrt{148} 2sqrt{37} ]

The diameter of the circle is ( 2sqrt{37} ). Since the radius is half of the diameter:

[ r frac{2sqrt{37}}{2} sqrt{37} ]

Alternative Method: Using the Pythagorean Theorem

Alternatively, you can use the Pythagorean theorem to determine the radius. This method involves treating the radius as the hypotenuse of a right triangle formed by the sides parallel to the axes.

The x-side of the triangle is the absolute difference in the x-coordinates of the two endpoints:

[ 5 - (-7) 12 ]

The y-side of the triangle is the absolute difference in the y-coordinates of the two endpoints:

[ |1 - 3| 2 ]

Using the Pythagorean theorem:

[ text{radius}^2 12^2 2^2 144 4 148 ]

Therefore:

[ text{radius} sqrt{148} 2sqrt{37} 2 cdot 2 sqrt{37} / 2 sqrt{37} ]

Conclusion

By using both the midpoint formula and the Pythagorean theorem, we have determined that the center of the circle is at ((-1, 2)) and the radius is (sqrt{37}). These methods are not only useful in solving geometric problems but also provide a solid foundation for SEO by ensuring that content is clear, concise, and accurate.