Finding Point Coordinates Using Section Formula in Geometry

This problem involves finding the coordinates of a point that is obtained by successively dividing line segments in given ratios. We will use the section formula for internal division to solve this.

Understanding the Section Formula

The section formula is a key concept in coordinate geometry. If a point \(P(x, y)\) divides the line segment joining points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) internally in the ratio \(m:n\), then the coordinates of \(P\) are given by:

\[ \left( \dfrac{mx_2 + nx_1}{m+n}, \dfrac{my_2 + nx_1}{m+n} \right) \]

In this problem, we have a sequence of divisions:

• p1 divides A1A2 in the ratio 1:1.
• p2 divides p1A3 in the ratio 1:2.
• p3 divides p2A4 in the ratio 1:3.

Step-by-Step Calculation of Coordinates

Step 1: Find the coordinates of p1

p1 divides A1 \((x_1, y_1)\) and A2 \((x_2, y_2)\) in the ratio 1:1. This means p1 is the midpoint of A1A2. Using the section formula with \(m=1\) and \(n=1\):

Coordinates of p1 \((x_{p1}, y_{p1})\) are:

\[ x_{p1} = \dfrac{1 \cdot x_2 + 1 \cdot x_1}{1+1} = \dfrac{x_1+x_2}{2} \]

\[ y_{p1} = \dfrac{1 \cdot y_2 + 1 \cdot y_1}{1+1} = \dfrac{y_1+y_2}{2} \]

So, p1 has coordinates \(\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)\).

Step 2: Find the coordinates of p2

p2 divides p1 \(\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)\) and A3 \((x_3, y_3)\) in the ratio 1:2. Here, the first point is p1 with coordinates \(\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)\), the second point is A3 with coordinates \((x_3, y_3)\), and the ratio is 1:2 (so \(m=1\) and \(n=2\)).

Coordinates of p2 \((x_{p2}, y_{p2})\) are:

\[ x_{p2} = \dfrac{1 \cdot x_3 + 2 \cdot \left(\dfrac{x_1+x_2}{2}\right)}{1+2} = \dfrac{x_3 + (x_1+x_2)}{3} = \dfrac{x_1+x_2+x_3}{3} \]

\[ y_{p2} = \dfrac{1 \cdot y_3 + 2 \cdot \left(\dfrac{y_1+y_2}{2}\right)}{1+2} = \dfrac{y_3 + (y_1+y_2)}{3} = \dfrac{y_1+y_2+y_3}{3} \]

So, p2 has coordinates \(\left(\dfrac{x_1+x_2+x_3}{3}, \dfrac{y_1+y_2+y_3}{3}\right)\).

Step 3: Find the coordinates of p3

p3 divides p2 \(\left(\dfrac{x_1+x_2+x_3}{3}, \dfrac{y_1+y_2+y_3}{3}\right)\) and A4 \((x_4, y_4)\) in the ratio 1:3. Here, the first point is p2 with coordinates \(\left(\dfrac{x_1+x_2+x_3}{3}, \dfrac{y_1+y_2+y_3}{3}\right)\), the second point is A4 with coordinates \((x_4, y_4)\), and the ratio is 1:3 (so \(m=1\) and \(n=3\)).

Coordinates of p3 \((x_{p3}, y_{p3})\) are:

\[ x_{p3} = \dfrac{1 \cdot x_4 + 3 \cdot \left(\dfrac{x_1+x_2+x_3}{3}\right)}{1+3} = \dfrac{x_4 + (x_1+x_2+x_3)}{4} = \dfrac{x_1+x_2+x_3+x_4}{4} \]

\[ y_{p3} = \dfrac{1 \cdot y_4 + 3 \cdot \left(\dfrac{y_1+y_2+y_3}{3}\right)}{1+3} = \dfrac{y_4 + (y_1+y_2+y_3)}{4} = \dfrac{y_1+y_2+y_3+y_4}{4} \]

So, p3 has coordinates \(\left(\dfrac{x_1+x_2+x_3+x_4}{4}, \dfrac{y_1+y_2+y_3+y_4}{4}\right)\).

Final Coordinates of p3

Based on our calculations, the coordinates of point p3 are \(\left(\dfrac{x_1+x_2+x_3+x_4}{4}, \dfrac{y_1+y_2+y_3+y_4}{4}\right)\). This matches the form of the first option.

Revision Table: Key Formulas

Additional Information: Centroid Connection

The formula for the coordinates of the centroid of a triangle with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) is \(\left(\dfrac{x_1+x_2+x_3}{3}, \dfrac{y_1+y_2+y_3}{3}\right)\). Notice that the coordinates of p2 we calculated are the centroid of the triangle formed by A1, A2, and A3.

Similarly, if we were to consider the point dividing the line joining the centroid of a triangle (formed by 3 points) and a fourth point in a specific ratio, we might get results similar to the centroid of a tetrahedron (formed by 4 points). The centroid of a tetrahedron with vertices \((x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3), (x_4, y_4, z_4)\) is \(\left(\dfrac{x_1+x_2+x_3+x_4}{4}, \dfrac{y_1+y_2+y_3+y_4}{4}, \dfrac{z_1+z_2+z_3+z_4}{4}\right)\). Our result for p3 in 2D matches the x and y coordinates of the centroid of a 4-point system if we were in 3D, or the "centroid" of the four points in 2D.

This problem shows a method to construct a point that can be interpreted as the "average" position or centroid of the four given points through a specific sequence of divisions.