Problem 241 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

The sides of a quadrilateral with an inscribed circle are \(7,10,5\) and \(2 \mathrm{~cm}\). and the sum of a pair of opposite angles is \(120^{\circ}\). Find area and radius of inscribed circle.

Step-by-Step Solution

Answer

Area of the quadrilateral is \(12 \,cm^2\) and the radius of the inscribed circle is \(1 \,cm\).

Step 1: Calculate Semi-Perimeter

In order to find the area of the quadrilateral using Brahmagupta's formula and the radius of the inscribed circle, we need to calculate the semi-perimeter. The semi-perimeter, denoted usually by \(s\), is equal to half the sum of the lengths of the sides of the quadrilateral. Hence, \(s = \frac{7 + 10 + 5 + 2}{2} = 12 \,cm\)

Step 2: Calculate Area Using Brahmagupta's Formula

Brahmagupta's formula gives the area \(K\) of the quadrilateral as follows: \(K = \sqrt{(s-a)(s-b)(s-c)(s-d)}\) where \(a, b, c, d\) are the sides of the quadrilateral and \(s\) is the semi-perimeter. Substituting our values, we have \(K = \sqrt{(12-7)(12-10)(12-5)(12-2)} = 12 \,cm^2\)

Step 3: Calculate Radius of Inscribed Circle

The formula for radius of the inscribed circle is \(r = \frac{K}{s}\) where \(K\) is area of the quadrilateral and \(s\) is the semi-perimeter. Hence, we substitute our values to get \(r = \frac{12}{12} = 1 \,cm\)

Key Concepts

Quadrilateral Inscribed Circle

A quadrilateral with an inscribed circle, also known as a cyclic quadrilateral, exhibits the elegant characteristic where a single circle touches all four sides. This type of quadrilateral has a special property: the sum of a pair of opposite angles equals 180 degrees. This fundamental characteristic can be crucial in solving problems, such as computing the area or finding other attributes like the radius of the inscribed circle.

Understanding the relationship between the inscribed circle and the quadrilateral is essential. Each point where the circle touches a side is called the point of tangency. The distance from any vertex to its corresponding point of tangency is fundamentally linked to the circle's radius and the quadrilateral's side lengths. This aspect becomes particularly pertinent when you wish to delve into more advanced geometry and solve complex problems involving cyclic quadrilaterals.

Semi-Perimeter of a Quadrilateral

The semi-perimeter of a quadrilateral is a key element in geometric calculations, particularly when using Brahmagupta's formula for finding an area. It is defined as half the sum of all side lengths of the quadrilateral. Mathematically, if a quadrilateral has sides of length 'a', 'b', 'c', and 'd', the semi-perimeter 's' is represented by the equation: \[ s = \frac{a + b + c + d}{2} \]This step simplifies further computations, such as finding the area or the radius of an inscribed circle. It is a crucial starting point for solving various problems and proves particularly useful in quadrilaterals with an inscribed circle, as showcased in the given exercise. Understanding the semi-perimeter concept is vital, as it directly influences the calculation of the quadrilateral's area and the extent to which the inscribed circle fits within it.

Area of a Quadrilateral

The area of a quadrilateral can be found in numerous ways, depending on the specific type of quadrilateral and the information available. For cyclic quadrilaterals, which have an inscribed circle, Brahmagupta's formula is exceptionally useful. The area 'K' is given by the formula: \[ K = \sqrt{(s-a)(s-b)(s-c)(s-d)} \]Where 's' is the semi-perimeter and 'a', 'b', 'c', 'd' are the lengths of the sides. This elegant formula encapsulates the relationship between the sides and the semi-perimeter, providing a direct means to compute the area. Whether dealing with homework problems or real-world applications, grasping this formula is essential for anyone studying mathematics, particularly geometry. By inserting the semi-perimeter and side lengths into this formula, as demonstrated in the exercise, the area can be quickly and accurately determined.