Problem 29 - Algebra 2

Question

GEOMETRY A circle has a radius with endpoints at \((2,5)\) and \((-1,-4)\) . Find the circumference and area of the circle.

Step-by-Step Solution

Answer

Radius = \(3\sqrt{10}\), circumference = \(6\pi\sqrt{10}\), area = \(90\pi\).

Step 1: Calculate the Length of the Radius

Given the radius has endpoints at \((2,5)\) and \((-1,-4)\), use the distance formula to calculate the length of the radius: \(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substituting the values, \(r = \sqrt{(-1 - 2)^2 + (-4 - 5)^2} = \sqrt{(-3)^2 + (-9)^2} = \sqrt{9 + 81} = \sqrt{90}\), which can be simplified to \(3\sqrt{10}\).

Step 2: Calculate the Circumference of the Circle

The circumference \(C\) of a circle is given by the formula \(C = 2\pi r\). Using \(r = 3\sqrt{10}\), the circumference is \(C = 2 \pi (3\sqrt{10}) = 6\pi \sqrt{10}\).

Step 3: Calculate the Area of the Circle

The area \(A\) of a circle is calculated using the formula \(A = \pi r^2\). Substituting \(r = 3\sqrt{10}\), we find \(A = \pi (3\sqrt{10})^2 = \pi \times 90 = 90\pi\).

Key Concepts

Distance Formula

The distance formula helps in finding the distance between two points in a plane. It's like using a ruler on a graph to see how far apart the points are. When you have two points,

• Point 1: \(x_1, y_1\)
• Point 2: \(x_2, y_2\)

the distance formula is written as \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). This formula is derived from the Pythagorean theorem.

Imagine a right triangle formed by horizontal and vertical lines that connect the two points. The distance is the hypotenuse of that triangle. Applying it to our example, we have endpoints \( (2,5) \) and \( (-1,-4) \).

Substituting these into the formula, we calculate the radius of the circle as \( r = \sqrt{(-1 - 2)^2 + (-4 - 5)^2} = \sqrt{(-3)^2 + (-9)^2} = \sqrt{9 + 81} = \sqrt{90}\). This simplifies to \(3\sqrt{10}\), the length of the radius in simplest form, helping us in further calculations.

Circumference of a Circle

The circumference is the distance around a circle, similar to the perimeter for polygons. To find this, we use the formula \(C = 2\pi r\), where \(r\) is the radius and \(\pi\) is approximately 3.14159.

In our example, with a radius \(3\sqrt{10}\), the calculation becomes \(C = 2 \pi (3\sqrt{10})\). Expanding this, we have \(C = 6\pi \sqrt{10}\). It’s a simple multiplication that tells us how long a string would be if wrapped around the circle's edge.

Understanding circumference also provides insight into circular motion and various everyday scenarios involving wheels or circular tracks.

Area of a Circle

The area of a circle is a measure of the surface covered by the circle. The formula \(A = \pi r^2\) is used, where \(\pi\) (Pi) remains a constant, and \(r^2\) is the radius squared.

For our circle with radius \(3\sqrt{10}\), the calculation steps are:

• First, square the radius: \( (3\sqrt{10})^2 = 9 \times 10 = 90\)
• Then multiply by \(\pi\): \(A = \pi \times 90 = 90\pi\)

The unit square of the area helps in practical applications like calculating materials needed for circular objects or understanding space coverage, such as pools or gardens.

Start thinking about the area as the amount of pizza topping on a round pizza, stretching from the center to the edge.