Problem 84 - College Algebra and Calculus: An Applied Approach

Question

Find the center and radius of the circle given the endpoints of the diameter of the circle.

Step-by-Step Solution

Answer

The center of the circle is the midpoint \( \left( \frac{{x1 + x2}}{2}, \frac{{y1 + y2}}{2} \right) \) and the radius is half the distance \( \frac{ \sqrt{{(x2 - x1)^2 + (y2 - y1)^2}} }{2} \).

Step 1: Determine the Coordinates of the Diameter

Let the coordinates of the endpoints of the diameter be given as (x1, y1) and (x2, y2). These coordinates will be used in our formula calculations.

Step 2: Find the Midpoint

The midpoint of two points, also the center of the circle, is found by averaging the x-coordinates and y-coordinates of those two points respectively. The midpoint M is given by \(M = \left( \frac{{x1 + x2}}{2}, \frac{{y1 + y2}}{2} \right) \).

Step 3: Calculate the Distance

The distance between two points in a plane is found by applying the distance formula which is based on the Pythagorean theorem. The distance D is calculated as \(D = \sqrt{{(x2 - x1)^2 + (y2 - y1)^2}} \).

Step 4: Compute the Radius

Knowing that the radius of the circle is half the length of the diameter, we divide the distance calculated in the previous step by 2. So, the radius R is given by \(R = \frac{D}{2} \).

Key Concepts

Understanding the Midpoint Formula

The midpoint formula is a useful tool in geometry, specifically in calculating the center of a circle when given the endpoints of its diameter. To find the midpoint, you'll need to average both the x-coordinates and the y-coordinates of the two given points. This process essentially gives you the location right between the two points on the coordinate plane.

This is important because the midpoint of the diameter is the center of the circle, a key component when discussing circle geometry. So, using the formula, the midpoint \( M \) is calculated as:

• The x-coordinate: \( \frac{{x_1 + x_2}}{2} \)
• The y-coordinate: \( \frac{{y_1 + y_2}}{2} \)

The resulting point \( M \) represents the center of the circle. Remember, visualizing this can help. Imagine a line segment with the endpoints as the tips, the midpoint formula tells you where the halfway mark is. This knowledge is pivotal for further calculations in circle problems.

Exploring the Distance Formula

The distance formula is derived from the Pythagorean theorem and offers a way to find the length of a segment on a coordinate plane. When working with circles, it helps to determine the diameter when endpoints are known. Essentially, it calculates how far apart two points are from each other.

• Formula: \( D = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}} \)

Through this formula, you compute the distance \( D \), which, when dealing with circles, is equivalent to the diameter when using the endpoints provided in the exercise. This step is crucial because knowing the distance allows you to further derive the properties of the circle, leading us to calculate its radius. Keep in mind, this formula measures the actual spatial gap between two points, making it widely applicable beyond just circle geometry.

The Importance of Radius Calculation

Knowing how to calculate the radius of a circle is essential for many geometric applications, including figuring out dimensions and solving problems related to circumference and area. When you have the diameter of a circle, derived from the distance formula, you can quickly find the radius by simply dividing by two.

• Radius formula: \( R = \frac{D}{2} \)

This step is fairly straightforward but is integral to understanding circles as the radius is often a common measurement needed in further calculations, such as finding the circumference \( C = 2\pi R \) and area \( A = \pi R^2 \). The radius essentially acts as the 'heartbeat' of the circle, and many properties revolve around it. By mastering the radius calculation, you can tackle a wide array of problems in geometry.