Question

Let \(O(0,0), P(3,4), Q(6,0)\) be the vertices of the triangle \(O P Q\). The point \(R\) inside the triangle \(O P Q\) is such that the triangles \(O P R, P Q R, O Q R\) are of equal area. The coordinates of \(R\) are (A) \(\left(\frac{4}{3}, 3\right)\) (B) \(\left(3, \frac{2}{3}\right)\) (C) \(\left(3, \frac{4}{3}\right)\) (D) \(\left(\frac{4}{3}, \frac{2}{3}\right)\)

Short answer

The coordinates of point R are (3, \(\frac{4}{3}\)).

Step-by-step solution

Understand the Problem

We have a triangle with vertices at points O(0,0), P(3,4), and Q(6,0). We need to find a point R inside this triangle such that triangles OPR, PQR, and OQR all have equal areas.

Express the area of each sub-triangle

Since the areas of the sub-triangles are equal, the area of triangle OPQ is three times the area of one of the sub-triangles. We can express the area of triangle OPQ using the formula for the area of a triangle A = \(\frac{1}{2} base \times height\). In this case, the base is OQ which has length 6 and the height is the y-coordinate of point P, which is 4. So, \( A_{OPQ} = \frac{1}{2} \times 6 \times 4 = 12\). Then, each sub-triangle's area is \(\frac{A_{OPQ}}{3} = \frac{12}{3} = 4\).

Determine the coordinates of point R

Since the areas of the sub-triangles are the same, point R must be located such that it's the centroid of triangle OPQ, which divides the triangle into three triangles of equal areas. The centroid of a triangle is given by the average of the x-coordinates and y-coordinates of its vertices. Thus, the x-coordinate of R is \(\frac{0+3+6}{3} = \frac{9}{3}=3\) and the y-coordinate of R is \(\frac{0+4+0}{3} = \frac{4}{3}\).

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves studying geometry using a coordinate system. This form of geometry allows us to describe the position of points, lines, and shapes in a two-dimensional plane by using ordered pairs of numbers called coordinates.

In the context of the given problem, we use a two-dimensional Cartesian coordinate system with x and y-axes to identify the positions of the vertices of a triangle. The vertices are given as points with coordinates, for example, O(0,0), P(3,4), and Q(6,0). This notation denotes that point O is at the origin of the coordinate system, point P is 3 units to the right and 4 units up from the origin, and point Q is 6 units to the right of the origin on the x-axis.

Calculating the centroid (the point where the medians of the triangle intersect) is an application of coordinate geometry. We calculate it by averaging the x-coordinates and the y-coordinates of the vertices of the triangle. This fundamental process is crucial for solving problems involving balance points or centers of mass in various geometrical shapes within the Cartesian plane.

Area of Triangle

Understanding how to calculate the area of a triangle is vital in both basic and advanced mathematics. The most common formula is \( A = \frac{1}{2} \times base \times height \), where the base is the length of one side of the triangle, and the height is the perpendicular distance from the chosen base to the opposite vertex.

For coordinate geometry problems, we often use the determinant method to find the area when the vertices are known. The formula involving the coordinates of the vertices \(A, B, \text{and} C\) is given by \( A_{\triangle ABC} = \frac{1}{2} |x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A -y_B)| \). This method provides an efficient way to work with triangles on the Cartesian plane without the need to measure physical lengths or heights.

In the problem given, the concept of the area is extended to find a point inside the triangle that creates three smaller triangles of equal area. This deepens our understanding of how the attributes of a shape, such as its balance point or centroid, relate to its area—a concept integral to solving sophisticated geometrical problems.

JEE Advanced Mathematics

The Joint Entrance Examination (JEE) Advanced is a highly competitive test for students in India aspiring to enter top-tier engineering institutes. Mathematics is a critical component of this examination, and it requires a thorough understanding of concepts including algebra, calculus, and geometry.

Problems similar to the one discussed involving centroids and areas of triangles are common in JEE Advanced mathematics. They test a student's ability to combine various mathematical concepts into a cohesive solution. This particular problem demands knowledge of coordinate geometry and understanding the properties of triangles, which are taught in the JEE Advanced syllabus.

Conquering such problems requires not only memorizing formulas but also grasping their underlying principles. Students are expected to visualize geometric configurations, manipulate algebraic expressions, and apply the relevant theorems to arrive at correct solutions. Mastery of these skills is essential for success in the JEE Advanced exam, making practice on such integrative problems an indispensable part of preparation for students.