Question

If 2,5 and 8 units are the radii of three circles, which touch each other, then the area of the triangle formed by joining the centres of these circles is (a) \(10 \sqrt{3}\) (b) \(20 \sqrt{3}\) (c) \(5 \sqrt{3}\) (d) \(12 \sqrt{3}\)

Short answer

Answer: \(20\sqrt{3}\)

Step-by-step solution

Find the side lengths of the triangle.

Two circles touch each other if the distance between their centers is equal to the sum of their radii. Thus, the lengths of the sides of the triangle formed by the centers of the circles are: - Side A (joining centers of circles with radii 2 and 5): 2 + 5 = 7 - Side B (joining centers of circles with radii 2 and 8): 2 + 8 = 10 - Side C (joining centers of circles with radii 5 and 8): 5 + 8 = 13

Calculate the semi-perimeter.

The semi-perimeter (denoted by s) of the triangle is half of its perimeter. Thus: s = (Side A + Side B + Side C) / 2 = (7 + 10 + 13) / 2 = 15

Use Heron's formula to find the area of the triangle.

Heron's formula states that the area of a triangle is equal to the square root of the product of the semi-perimeter (s) and the distance between s and each side (s - A, s - B, s - C). Therefore, the area of the triangle is: Area = \(\sqrt{s(s - A)(s - B)(s - C)} = \sqrt{15(15 - 7)(15 - 10)(15 - 13)} = \sqrt{15 \cdot 8 \cdot 5 \cdot 2}\) After simplifying the expression inside the square root, we find the area: Area = \(\sqrt{15 \cdot 8 \cdot 5 \cdot 2} = \sqrt{1200} = 20\sqrt{3}\) So, the area of the triangle formed by joining the centers of the circles is \(20\sqrt{3}\), which corresponds to option (b).

Key concepts

Triangle Area Calculation

Calculating the area of a triangle is a fundamental task in geometry. One of the well-known methods is Heron's formula, a remarkable algorithm that only requires the lengths of the three sides of the triangle, making it especially useful for cases where the height is difficult to determine. Heron's formula states that the area of a triangle with sides of lengths 'a,' 'b,' and 'c' can be found using the semi-perimeter 's' where:
\(s = \frac{a + b + c}{2}\)
Then, the area 'A' is given by the square root of the product of 's' and the difference of 's' and each side:
\(A = \sqrt{s(s - a)(s - b)(s - c)}\)
This formula is derived from the properties of triangles in Euclidean geometry and is applicable to all types of triangles, whether they are acute, obtuse, or right-angled. To ensure clarity for students, it is vital to provide a step-by-step calculation, breaking down the process into semi-perimeter calculation followed by the application of Heron's formula.

Geometry in Mathematics

Geometry, the branch of mathematics that deals with shapes, sizes, and properties of space figures, is integral to understanding the world around us. Triangles, specifically, are among the simplest yet most studied polygons due to their fascinating properties and applications. From the Pythagorean theorem for right-angled triangles to the various triangle congruence criteria, triangles are cornerstones in the study of geometry. A deep inquiry into triangles often involves examining triangles' sides, angles, and relevant theorems to solve complex problems, much like the use of Heron's formula for area calculation. Understanding these principles is instrumental for students not only to solve geometric problems but also to apply them in fields such as engineering, architecture, and various sciences where spatial reasoning is essential.

Circle Properties

Circles are another fundamental concept in geometry characterized by their symmetry and uniform shape. A circle is defined as a set of points in a plane that are all equidistant from a fixed point called the center. The distance from the center to a point on the circle is called the radius, and it is a crucial dimension for various circle properties and calculations, including the area and the circumference. In the context of the exercise, the circle properties discussing how circles can touch each other - tangency - are used. When circles are externally tangent, the distance between their centers is equal to the sum of their radii. This relationship is pivotal in establishing the side lengths of a triangle formed by joining the centers of three mutually tangent circles, which is precisely what the problem at hand demonstrates. Explaining these concepts lucidly can significantly aid students in grasping the geometric principles involved.