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2. Section formula using graphical representation

The section formula is used to get the coordinates of a point by splitting a line segment into two parts with the given ratio.

Let us look at the graph carefully.

From the graph, the line \(AB\) is divided at \(P\) in the ratio \(m : n\).

Therefore,

\frac{AP}{PB} = \frac{m}{n}

So, \(A'P' : P'B'\) is also \(m : n\).

\frac{A^{'} P^{'}}{P^{'} B^{'}} = \frac{m}{n}

\(n(A'P') =\) \(m(A'B')\)

\(n(x - x_1) =\) \(m(x_2 - x)\)

\(nx - nx_1 = mx_2 - mx\)

\(mx + nx = mx_2 + nx_1\)

\(x(m + n) = mx_2 + nx_1\)

x = \frac{m x_{2} + n x_{1}}{m + n}

Similarly,

y = \frac{m y_{2} + n y_{1}}{m + n}

1. The coordinates of the point \(P(x, y)\) which divides the line segment joining the points \(A(x_1, y_1)\) and \(B(x_2, y_2)\), internally in the ratio \(m : n\) are:

P (x, y) = (\frac{m x_{2} + n x_{1}}{m + n}, \frac{m y_{2} + n y_{1}}{m + n})

This is known as the section formula.

2. If the point \(P(x, y)\) divides the line segment in the ratio \(k : 1\), then the coordinates are:

P (x, y) = (\frac{k x_{2} + x_{1}}{k + 1}, \frac{k y_{2} + y_{1}}{k + 1})

3. If the point \(P(x, y)\) divides the line segment in the ratio \(1 : 1\), then the coordinates are:

P (x, y) = (\frac{1 \times x_{2} + 1 \times x_{1}}{1 + 1}, \frac{1 \times y_{2} + 1 \times y_{1}}{1 + 1}) = (\frac{x_{2} + x_{1}}{2}, \frac{y_{2} + y_{1}}{2})

P (x, y) = (\frac{x_{2} + x_{1}}{2}, \frac{y_{2} + y_{1}}{2})

This is known as the mid-point formula of the line segment.

4. If \(A(x_1, y_1)\), \(B(x_2, y_2)\) and \(C(x_3, y_3)\) be the vertices of a triangle, then the centroid of a triangle is:

G (x, y) = (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3})

Important!

The point at which all the \(3\) medians intersect is a centroid.

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2. Section formula using graphical representation

Theory: