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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

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