Introduction to Cyclic Trapezoids

A trapezoid inscribed in a circle, known as a cyclic trapezoid, is a fascinating geometric figure with unique characteristics. Specifically, in a cyclic trapezoid, the opposite angles are supplementary, and this property allows for a variety of useful formulas to find attributes like the radius of the circumscribed circle.

In this article, we will delve into the process of finding the radius of the circumcircle of a cyclic trapezoid using different methods. Whether you’re a math enthusiast, a student, or a professional in a field requiring precise measurements, understanding these calculations can be incredibly valuable.

Understanding the Problem

When a trapezoid is inscribed in a circle, it is referred to as a cyclic trapezoid. For a cyclic trapezoid, the opposite angles are supplementary (i.e., their sum is 180°). Given the lengths of the bases and the height, we can determine the radius of the circumcircle. This article will guide you through a detailed step-by-step process to achieve this.

Steps to Find the Radius of the Circumcircle

Step 1: Identify the lengths of the bases and height

Let's denote the lengths of the two bases of the trapezoid as (mathbf{a}) and (mathbf{b}), where (a) is the longer base, and the height is (mathbf{h}).

Step 2: Use the relevant formula

There are two primary methods to find the radius (R) of the circumcircle of the cyclic trapezoid.

Method 1: Using the Bases and Height

According to the formula, if the angles between the non-parallel sides and the longer base are known, the radius (R) can be calculated as:

[ R frac{a cdot b}{2 sin theta} ]

Here, (theta) is the angle between one of the non-parallel sides and the longer base.

Method 2: Using the Legs and Height

If the lengths of the non-parallel sides (c) and (d) are known, another formula can be used:

[ R frac{1}{2} sqrt{frac{ab cdot cd - ac cdot bd ad cdot bc}{a cdot b^2}} ]

Step 3: Example Application

Consider a trapezoid with bases (a 10) and (b 6), and legs (c 5) and (d 7).

Using the second method:

[ R frac{1}{2} sqrt{frac{10 cdot 6 cdot 7 cdot 5 - 10 cdot 5 cdot 7 cdot 6 7 cdot 6 cdot 5 cdot 10}{10 cdot 6^2}} ]

Let's compute each term:

(ab cdot cd 60 cdot 35 2100) (ac cdot bd 50 cdot 42 2100) (ad cdot bc 70 cdot 30 2100) (a cdot b^2 10 cdot 6^2 360)

Plugging these values into the formula:

[ R frac{1}{2} sqrt{frac{2100 - 2100 2100}{360}} frac{1}{2} sqrt{frac{2100}{360}} frac{1}{2} sqrt{frac{35}{6}} approx 2.76 ]

Visual Explanation of the Problem

Consider a geometric diagram of a cyclic trapezoid. Here are some key observations and calculations:

The height of the trapezoid is twice the radius, i.e., (2R). The length of (CE 2R). Since the trapezoid is isosceles, (AC DB), so (AE frac{18 - 6}{2} 6). Since (ACE) is a right triangle, (tan(angle ACE) frac{2R}{6} frac{R}{3}). Considering quadrilateral (OFGD), and knowing that (angle OFD frac{1}{2} angle ACE), it follows that (tan(frac{1}{2} theta) frac{3}{R}).

By using the double angle formula for tangents, we get:

[ frac{2 cdot 3}{1 - left(frac{3}{R}right)^2} frac{3}{R} ]

Solving this equation yields (R 3sqrt{3}).

Conclusion

Understanding and applying the correct geometric formulas can help in solving problems related to cyclic trapezoids and their circumcircles. Whether you are solving a theoretical problem or a practical engineering challenge, these methods provide a solid foundation for accurate measurements and calculations.