Understanding the Radius of a Circle Inscribed in a Tangential Quadrilateral

Tangential quadrilaterals are fascinating geometric shapes, and understanding their properties can enhance our problem-solving skills in mensuration and geometry. A circle with center O is inscribed in a tangential quadrilateral ABCD, meaning that all four sides of the quadrilateral are tangent to the circle. This article delves into the properties of such quadrilaterals, particularly focusing on the calculation of the radius of the inscribed circle.

What is a Tangential Quadrilateral?

A tangential quadrilateral is a convex quadrilateral whose all sides are tangent to a single circle within itself. This circle is known as the incircle or inscribed circle, and its center and radius are referred to as the incenter and inradius(), respectively.

Properties of Tangential Quadrilaterals

In a tangential quadrilateral, the sum of the lengths of two opposite sides is equal. More specifically, if the side lengths are a, b, c, and d, then we can write:

a c b d

Additionally, the area of a tangential quadrilateral can be found using the following formula:

Area √abcd

Calculation of the Inradius

Given a tangential quadrilateral with side lengths as 18 cm, 16 cm, 10 cm, and 12 cm, we can use the formula to calculate the inradius (r) of the inscribed circle:

Area of the quadrilateral ABCD √(18 × 16 × 10 × 12)

√(32 × 2 × 2 × 2 × 5 × 2 × 2 × 3 × 3 × 2)

√(22 × 22 × 22 × 22 × 32 × 3 × 5)

48√15 cm2

However, the area can also be expressed in terms of the inradius:

Area 28r cm2

Setting the two expressions for the area equal to each other:

28r 48√15

Solving for r:

r 48√15 / 28

r 12√15 / 7 ≈ 6.6392 cm

Conclusion

The radius of the circle inscribed in a tangential quadrilateral can be accurately calculated using the properties outlined in this article. Understanding these properties not only enhances our geometric knowledge but also aids in solving complex problems in mensuration. The inradius gives us significant information about the size and shape of the tangential quadrilateral, making it a valuable concept in both theoretical and practical applications.

Photo of Large Scale Drawing: My large-scale drawing with a calculated radius of approximately 6.64 cm, not a typo.