SECTION FORMULA

1. Internal division

Consider a point P(x, y) divides the line segment joining A(x₁, y₁) and B(x₂, y₂) in the ratio m : n.

Draw $AM \bot \,OX$

$PN \bot \,OX$

$BL \bot \,OX$

$AQ \bot PN$

$PR \bot \,BL$

$\Delta APQ\, \sim \,\Delta PBR$

$\therefore \,\frac{{AQ}}{{PR}} = \frac{{PQ}}{{BR}} = \frac{{AP}}{{PB}}$

$\Rightarrow \,\,\frac{{x - {x_1}}}{{{x_2} - x}} = \frac{{y - {y_1}}}{{{y_2} - y}} = \frac{m}{n}$

$\,\frac{{x - {x_1}}}{{{x_2} - x}} = \frac{m}{n}$
$\Rightarrow \,\,nx - n{x_1} = m{x_2} - mx$
$\Rightarrow \,mx + nx = m{x_2} + n{x_1}$
$\Rightarrow \,\,x = \frac{{m{x_2} + n{x_1}}}{{m + n}}$

$\,\frac{{y - {y_1}}}{{{y_2} - y}} = \frac{m}{n}$
$\Rightarrow \,\,ny\,\, - \,n{y_1} = m{y_2} - my$
$\Rightarrow \,\,my\, + \,ny\, = \,m{y_2} + n{y_1}$
$\Rightarrow \,y = \frac{{m{y_2} + n{y_1}}}{{m + n}}\,$

The coordinates of point P are $\left( {\frac{{m{x_2} + n{x_1}}}{{m + n}},\frac{{m{y_2} + n{y_1}}}{{m + n}}} \right)$

Example: Find the coordinates of point P which divides the line segment joining A(5, 3) and B(2, 1) in the ratio 5 : 2.

Solution:

$P = \left( {\frac{{m{x_2} + n{x_1}}}{{m + n}},\,\frac{{m{y_2}\, + \,n{y_1}}}{{m + n}}} \right)$

$P = \,\left( {\frac{{5 \times 2 + 2 \times 5}}{{5 + 2}},\,\frac{{5 \times 1 + 2 \times 3}}{{5 + 2}}} \right)$

$P = \,\left( {\frac{{20}}{7},\,\frac{{11}}{7}} \right)$

2. External division

Consider a point P(x, y) divides the line segment joining A(x₁, y₁) and B(x₂, y₂) externally in the ratio m : n such that AP : BP = m : n.

Draw $AM \bot OX$

$\begin{array}{l} BN \bot \,OX\\ PL \bot OX\\ AQ \bot PL\\ BR \bot PL \end{array}$

$\,\frac{{x - {x_1}}}{{x - {x_2}}} = \frac{m}{n}$
$\Rightarrow \,nx - n{x_1} = mx - m{x_2}$
$\Rightarrow \,mx - nx = m{x_2} - n{x_1}$
$\Rightarrow \,x = \frac{{m{x_2} - n{x_1}}}{{m - n}}$

$\,\frac{{y - {y_1}}}{{y - {y_2}}} = \frac{m}{n}$
$\Rightarrow \,ny - n{y_1} = my - m{y_2}$
$\Rightarrow \,my - ny = m{y_2} - n{y_1}$
$\Rightarrow \,y = \frac{{m{y_2} - n{y_1}}}{{m - n}}$

The coordinates of point P are $\left( {\frac{{m{x_2} - n{x_1}}}{{m - n}},\frac{{m{y_2} - n{y_1}}}{{m - n}}} \right)$ .

Example:

Find the coordinates of point which divides the line joining A(5, 2) and B(–1, 2) externally in the ratio 3 : 1.

Solution:

$P = \left( {\frac{{m{x_2} - n{x_1}}}{{m - n}},\frac{{m{y_2} - n{y_1}}}{{m - n}}} \right)$

$P\, = \,\left( {\frac{{3 \times - 1 - 1 \times 5}}{{3 - 1}} - \frac{{3 \times 2 - 1 \times 2}}{{3 - 1}}} \right)$

$P\, = \,\left( {\frac{{ - 8}}{2},\frac{4}{2}} \right)$

$P = \left( { - 4,2} \right)$

3. Centroid of a triangle

A Centroid of a triangle is the point where the three medians of the triangle meet. The centroid of the triangle is also called as the gravity of the triangle, which is shown as follows:

Here, G is the centroid of a triangle ABC.

4. Median of a triangle

The median of a triangle is a line segment joining a vertex and the midpoint of the opposite side of a triangle, which is shown as follows:

Here, ${m_a},{m_b},{\rm{ and }}{m_c}$ are the three medians intersect at a single point G.

Properties of Centroid of a triangle

Centroid is the centre of the object.
Centroid is also known as gravity, geo centre and bary centre.
Centroid of a triangle represents the interior of the triangle.

The formula for the Centroid of a triangle of points A, B and C is:

$\left( {\frac{{{x_1} + {x_2} + {x_3}}}{3},\frac{{{y_1} + {y_2} + {y_3}}}{3}} \right)$

Example:

Find the centroid coordinates of the triangle whose vertices are $\left( {2,4} \right),\left( {3,5} \right){\rm{ and }}\left( {1,0} \right)$ is determined as follows:

Solution

Consider the centroid formula:

$\frac{{{x_1} + {x_2} + {x_3}}}{3},\frac{{{y_1} + {y_2} + {y_3}}}{3}$

Let $\left( {{x_1},{y_1}} \right) = \left( {2,4} \right),\left( {{x_2},{y_2}} \right) = \left( {3,5} \right){\rm{ and }}\left( {{x_3},{y_3}} \right) = \left( {1,0} \right)$ , substitute the value as follows:

$\begin{array}{c} \left( {\frac{{2 + 3 + 1}}{3},\frac{{4 + 5 + 0}}{3}} \right) = \left( {\frac{6}{3},\frac{9}{3}} \right)\\ = \left( {2,3} \right) \end{array}$

Therefore, the centroid coordinates of the triangle is (2, 3).

Posted on December 28, 2014 by admin

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