Distance Formula

India is seventh-largest country and second in population in the world. It is a country of Asia. Delhi is the capital of India. There are a total of $28$ states and $8$ union territories in India at present. Each state has their capitals. Each state of India has an administrative, legislative and judicial capital some state. These functions are conducted in one capital.

Ravi plotted some capitals as points on a graph paper and named them as $A, B, C, .....$ etc. as shown below:

The list of the cities which are marked in the graph are given using the table:

The coordinates of a point which lies on the $x-$axis and equidistant from New Delhi and Hyderabad are

An ant is walking across a chessboard.

Its current position on the board can be represented by the point $A$ on the coordinate axes as:

The coordinates of the point on the line $y=5$ that will be closest to the ant is

India is seventh-largest country and second in population in the world. It is a country of Asia. Delhi is the capital of India. There are a total of $28$ states and $8$ union territories in India at present. Each state has their capitals. Each state of India has an administrative, legislative and judicial capital some state. These functions are conducted in one capital.

Ravi plotted some capitals as points on a graph paper and named them as $A, B, C, .....$ etc. as shown below:

The list of the cities which are marked in the graph are given using the table:

From the graph, the ratio of the distance between Ranchi-Patna and Gandhinagar-Mumbai is

The constellation of Pegasus is named after the winged horse in Greek mythology. According to the myth, when Perseus killed the Gorgon Medusa (whose hair was made of snakes), Pegasus was born from her blood.

In the constellation of Pegasus, the horse's body is represented by four stars called "The Great _____".

The stars at the corners of this quadrilateral can be drawn on a Cartesian plane at $A\left(-1, 5\right), B\left(3, 2\right), C\left(0, -2\right), D\left(-4, 1\right)$.

The length of side $AB$ of the quadrilateral is

Show that $(0,1), (1,2), (2,3)$and $(3,4)$ are collinear.

Show that $(0,2), (1,3), (2,4)$and $(3,5)$ are collinear.

Show that $(1,4), (2,5), (3,6)$and $(4,7)$ are collinear.

Show that $(0,4), (1,6), (2,8)$and $(3,10)$ are collinear.

Find $a$ if the following points are the vertices of a right triangle with $\angle A=90°$.

$A\left(2, 3\right), B\left(7, a\right), C\left(4, 7\right)$

Find $a$ if the following points are the vertices of a right triangle with $\angle A=90°$.

$A\left(4,3\right), B\left(6,-2\right), C\left(a,-3\right)$

Prove that the points $(1,3), (2,5)$ and $\left(3, 5\right)$ are on the same line.

The distance of the point $\left(-3.-4\right)$ from oringin is

Find the distance between the points $(1,5)$ and $(5,8)$.

The distance between the co-ordinates of a point $\left(p,q\right)$ from the origin is

Akhila says, "points $A(1,3), B(2,2), C(5,1)$ are collinear". Do you agree with Akhila?

Find the length of medians of a triangle whose vertices are $(1,-1),(0,4)$ and $(-5,3)$.

If $(0,0)$ and $(3,\sqrt{3})$ are two vertices of an equilateral triangle, then find third vertex.

Using a graph paper to answer the given questions.
(Take $2 cm=1$ unit on both axes)
Plot the points $A(-4, 2)$ and $B(2, 4)$.

Show that the points $\mathrm{A}(-5, 6), \mathrm{B}(3, 0)$ and $\mathrm{C}(9, 8)$ are the vertices of an isosceles right-angled triangle. If area of triangle is $k \mathrm{sq}.\mathrm{units}$ find the value of $k$.

Draw quadrilateral, if its vertices are $\left(-2,-2\right),\left(-4,2\right),\left(-6,-2\right)$ and $\left(-4,-6\right)$. Also, mention type of obtained quadrilateral.