Section and Mid-Point Formulae — Part 1: The section formula: visual lesson, ICSE Class 10 Mathematics

Lesson

Section and mid-point formulae, Part 1

A point on a segment splits it into two parts. The section formula gives the coordinates of that point from the two ends and the ratio of the parts.

In this lesson you will

Explain what it means for a point to divide a segment internally in a given ratio
Find the point dividing a segment internally in a given ratio, including points of trisection
Find the ratio in which a given point divides a segment
Find the ratio in which an axis or a line divides a segment, and the point of division

segment AB

$P$ splits $AB$ into $2$ parts and $3$ parts: $AP : PB = 2 : 3$.

Concept

What you will be able to do

Explain internal division in a ratio $m_1 : m_2$
Use the section formula, including points of trisection
Find the ratio in which a given point divides a segment
Find where an axis or a line cuts a segment, and in what ratio

Concept

Dividing a segment internally

$P$ divides $AB$ internally in the ratio $m_1 : m_2$ when $P$ lies on the segment $AB$ and

$m_1$ goes with the part next to $A$ , $m_2$ with the part next to $B$
if $m_1 < m_2$ , $P$ is nearer to $A$
the ratio $1 : 1$ gives the mid-point

\dfrac{AP}{PB} = \dfrac{m_1}{m_2}

segment AB in three equal parts

$AP$ is $1$ part, $PB$ is $2$ parts: $P$ divides $AB$ internally in the ratio $1 : 2$.

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