This topic falls under “Coordinate Geometry“. We will learn the following topics today.

The Distance Formula (refer to the diagram as well)

Let the two given points be and

In the diagram, you can see that the is a right angled triangle and . Which means that (Pythagoras Theorem)

Given and

Hence

Therefore the distance between any two points and is

Notes: If a point is on , its ordinate is , therefore the point on is taken as . Similarly, if the point is on , its abscissa is , therefore the point on is taken as .

Three points are said to be co-linear if and only if (as shown in the diagram). Which means that the distance from plus the distance from is equal to the distance from .

Circumcentre of a Triangle

It is a point that is equidistant from the three vertices of a triangle. i.e. if point is equidistant from the three vertices , then of .

What this means is that if a circle is drawn with as the center and any of the vertices as the radius, the circle will touch all the three vertices of the .

Section Formula

This is used to find a point that divides a line segment joining two points in a given ratio.

Let be the line segment. Let coordinates of and .

Let be a point dividing in the ratio of .

We need to find .

Refer to the diagram.

Since and are similar

Note: If instead of you used , then the formula would become as follows:

Points of Trisection

Let points lie on a line segment such that it divides the line in three equal parts i.e.

Point divides the line segment in the ratio of

Similarly, point divides the line segment in the ratio

Midpoint Formula

Let points lie on a line segment such that it divides the line in two equal parts i.e.

Point divides the line segment in the ratio of

The centroid of a Triangle

The centroid of a triangle is the point of intersection of its medians and the centroid divides each of the medians in the ratio of .

To find the coordinates of the centroid:

First, find the coordinates of the midpoint of the sides of the triangle.

The find out the coordinates of the centroid between the vertex and the opposite midpoint.