We can easily measure the distance between two points using a meter scale. However, imagine you were not provided a scale and were asked to measure the distance between two points. In such cases, it is best to use the distance and section formula.
The distance formula is used to find the distance between two defined points on a graph (in the absence of a scale).
The distance between two points and is given by the formula
Plot and on the graph. Draw and perpendicular to the -axis. Also draw perpendicular to . Now, we observe that triangle forms a right-angled triangle. Then by the Pythagorean theorem,
From this proof we can derive the following corollary:
The distance of point from the origin is given by
Find the distance between the points and .We have
Find the distance of the point from the origin.We have
Find the coordinates of points on the -axis which are at a distance of units from the point .Let the co-ordinates of the point on the -axis be . Since, distance we have
Therefore, the required points on the -axis are and .
Find the point on the -axis which is equidistant from the points and .Let the required point on the -axis be Then the distance between and is equal to the distance between and
Therefore, the required point is .
Condition for three points to be collinear
Three points are said to be collinear if and only if the sum of distances between any two points is equal to distance between that point and third point. That means,
Verify whether the points and are collinear or not.We have
As, , the given points are collinear.
Sometimes we are given four points and asked to comment on the nature of the quadrilateral which is formed by joining them. For this, we have to recall the following:
A quadrilateral is a
- rectangle, if its opposite sides are equal and diagonals are equal;
- square, if all its sides are equal and diagonals are equal;
- parallelogram, if its opposite sides are equal;
- rhombus, if its sides are equal.
Show that the points are the vertices of an isosceles right-angled triangle. Also, find the area of the triangle.
Since triangle is a right-angled triangle.
Now, since triangle is an isosceles triangle.
Hence, triangle is a right-angled isosceles triangle.
The area of triangle is
Show that the points are the vertices of a rectangle. Also, find the area of the rectangle.We have
which implies and Now,
Thus, is a quadrilateral whose opposite sides are equal and the diagonals are equal, which implies that is a rectangle.
The area of quadrilateral is
Find the area of a circle which has its center at and passes through the point .The radius of the circle is equal to the distance between the points and
Therefore, the area of the circle is
Main article : Section Formula
The section formula gives the coordinates of a point which divides the line joining two points in a ratio, internally or externally.
divides the line joining two points and in the ratio internally, then the coordinates of are given byIf a point
If divides externally in the ration , then
Find the coordinates of point which divides the line joining and in the ratio .Let the coordinates of be Then