Problem 46 - Basic Technical Mathematics with Calculus

Question

In Exercises \(41-46,\) solve the given problems. In Exercises \(41-44\) refer to Fig. 4.42. In Exercises 41-43, express the other parts in terms of the listed parts, which are assumed known. Find the exact perimeter of an equilateral triangle is inscribed in a circle (each vertex is on the circle) of circumference \(20 \pi\)

Step-by-Step Solution

Answer

The exact perimeter of the triangle is \(30\sqrt{3}\).

Step 1: Understanding the relation between the circle and the triangle

The equilateral triangle is inscribed in a circle, meaning each vertex of the triangle touches the circle. The circle is the circumcircle of the triangle, and its radius is called the circumradius of the triangle.

Step 2: Determine the circle's radius

The circumference of the circle is given as \(20 \pi\). Recall that the formula for the circumference of a circle is \(2\pi R\), where \(R\) is the radius. Using the given circumference: \[ 2\pi R = 20 \pi \] Divide both sides by \(2\pi\): \[ R = \frac{20\pi}{2\pi} = 10 \] Thus, the circle's radius (circumradius) is \(10\).

Step 3: Use the relation between side length and circumradius

For an equilateral triangle, the relationship between the side length \(s\) and the circumradius \(R\) is given by the formula: \[ s = \sqrt{3}R \] Substitute the value of \(R = 10\): \[ s = \sqrt{3} \times 10 = 10\sqrt{3} \] So, the side length of the triangle is \(10\sqrt{3}\).

Step 4: Calculate the perimeter of the triangle

The perimeter \(P\) of an equilateral triangle with side length \(s\) is given by the formula: \[ P = 3s \] Substitute the value of \(s = 10\sqrt{3}\): \[ P = 3 \times 10\sqrt{3} = 30\sqrt{3} \] This is the exact perimeter of the triangle.

Key Concepts

Circumradius

In geometry, the circumradius is a crucial concept when dealing with inscribed figures, particularly when a polygon is circumscribed by a circle.The circumradius is the radius of that circle, and every vertex of the polygon touches the circle.

For an equilateral triangle, the circumradius is especially important because it is directly related to its side length.The circumradius helps simplify complex geometrical problems by providing a single measure that connects the circle and the polygon.When you have the radius of a circumcircle, you can easily solve for other properties of the polygon, like side lengths or perimeters.

• In our example, the circumradius is determined by the circle's circumference, which is given as \(20\pi\).
• Using the relative formula \(2\pi R = 20\pi\), we find \(R = 10\).

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are equal in length.This property also means that all three internal angles are equal, each measuring \(60^\circ\).

Equilateral triangles have unique relationships with circles, especially when inscribed or circumscribed.When an equilateral triangle is inscribed in a circle, the circumcircle, each vertex of the triangle lies on the circle's circumference.

• In such a setup, the circumradius aids in calculating the side length, using the formula \(s = \sqrt{3}R\).
• Thus, if the circumradius \(R\) is known, the side length \(s\) can be easily calculated, simplifying the process of further geometric calculations.

Perimeter Calculation

Calculating the perimeter of an equilateral triangle is straightforward once you know the side length.The perimeter \(P\) is simply three times the side length, thanks to the equality of all sides.

In our problem, after determining the side length as \(10\sqrt{3}\) using the circumradius, perimeter calculation is a small step away.

• Use the formula \(P=3s\) to find \(P=3 \times 10\sqrt{3} = 30\sqrt{3}\).
• The result is the exact perimeter, which makes it clear and concise to express.

Inscribed Figures

Inscribed figures are those shapes or polygons placed within a circle such that all vertices of the figure touch the circle's boundary. This relationship between the figure and the circle is valuable in many geometrical problems because it establishes a clear link between linear and circular measurements.

For an equilateral triangle inscribed within a circle, like in our problem, the circle becomes the triangle’s circumcircle.

• This scenario allows us to use the circumradius to find important properties, like the side length or the perimeter of the triangle.
• Understanding inscribed figures and their geometric properties allows these kinds of problems to be addressed efficiently, leveraging the symmetry and equality inherent in the shapes involved.