The in centre of a triangle whose vertices \(\mathrm{A}(2,4), \mathrm{B}(2,6)\) and \(\mathrm{C}(2+\sqrt{3}, 5)\) is.... (a) \([2+(1 / \sqrt{3}), 5]\) (b) \([1+\\{1 /(2 \sqrt{3})\\},(5 / 2)]\) (c) \((2,5)\) (d) None of these

Coordinate geometry, also known as analytic geometry, is a part of mathematics that involves studying geometry using a coordinate system. This approach combines algebra and geometry, allowing problems, such as the location of the incenter of a triangle, to be solved using algebraic equations.

For example, in the provided exercise, the coordinates of the vertices of a triangle are given. These coordinates represent points on the Cartesian plane, and by using coordinate geometry, you can determine the triangle's incenter. The incenter is the point where the angle bisectors of a triangle intersect, and it is also the center of the triangle's incircle - the circle inscribed within the triangle. Knowing how to find the incenter using the vertices' coordinates is a practical application of coordinate geometry and a typical problem that might appear in exams like the JEE.

The distance formula is a derived application of Pythagoras' theorem in coordinate geometry that calculates the distance between two points on the Cartesian plane. It is essential for solving a variety of geometry problems, including finding the lengths of a triangle's sides when given the coordinates of its vertices.

The formula is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In this problem, we compute the distances between the vertices of the triangle using this formula. These distances are used in further calculations, such as finding the triangle's incenter, which is crucial for certain problems on the JEE. Mastery of this formula not only helps in solving direct distance problems but also forms the foundation for other complex geometrical constructions.

Geometry problems for the Joint Entrance Examination (JEE), which is an engineering entrance assessment in India, often test a student's understanding of concepts like coordinate geometry and the distance formula. These problems require not only the knowledge of formulas but also the ability to interpret geometric concepts on a coordinate plane.

One such problem that might be encountered on the JEE is finding the incenter of a triangle, as seen in the exercise. JEE geometry problems often involve multiple steps and require a student to be comfortable with connecting various geometrical properties, such as angle bisectors, circumcenters, and incenters, with algebraic methods. The key to success is thorough practice and deep understanding of the relationship between geometry and algebra as demonstrated in the step-by-step solution to the problem about the incenter of a triangle.