Section Formula

The three vertices of a square $ABCD$ are $A\left(-1,1\right), B\left(3,1\right)$ and $C\left(3,-3\right).$ Find the coordinates of $D.$ Also find area of square.

Find the ratio in which the point $\left(11, 20\right)$ divides the line segment joining the points $\left(15, 5\right)$and $\left(9, 20\right)$.

If $P\left(\frac{a}{3}, 4\right)$ is the midpoint of the line segment joining the points $Q\left(-6, 5\right)$ and $R\left(-2, 3\right)$, then the value of $a$ is

If $p\left(x,y\right)$ is any point on the line passing through $A\left(x_1,y_1\right)$ and $B\left(x_2,y_2\right)$ then the ratio in which $\text{'}p\text{'}$ divides $\overline{AB}$ is

In the figure shown below, $\mathrm{DE} \parallel \mathrm{BC}$.

Find the coordinates of point $\mathrm{E}$.

Shown below is a map of Giri's neighbourhood.

Giri did a survey of his neighbourhood and collected the following information.
* The hotel, mall and the main gate of the garden lie in a straight line.
* The distance between the hotel and the mall is half the distance between the mall and the main gate of the garden.
* The bus stand is exactly midway between the main gate of the garden and the fire station.
* The mall, bus stand and the water tank lie in a straight line.

Giri proposed a plan to make a triangular pathway by joining the midpoints of the sides of the triangular garden.
What will be the area, in square units, enclosed by the triangular pathway?

Two statements are given below- one labelled Assertion $\left(A\right)$ and the other labelled Reason $\left(R\right)$. Read the statements carefully and choose the option that correctly describes statements $\left(A\right)$ and $\left(R\right)$.

Assertion $\left(A\right)$: The origin is the ONLY point equidistant from $\left(2, 3\right)\text{ and }\left(-2, -3\right)$.

Reason $\left(R\right)$: The origin is the midpoint of the line joining $\left(2, 3\right)\text{ and} \left(-2, -3\right)$

Assertion: The coordinates of $P\left(2,0\right)$ on $x-$axis is equidistant from point $A\left(-2,0\right)$ and $B\left(6,0\right).$

Reason: Since, $P\left(2,0\right)$ is the midpont of $A\left(-2,0\right)$ and $B\left(6,0\right).$

Find the ratio in which point $\left(-1, k\right)$ divides the line joining the points $\left(-3, 10\right)$ and $\left(6, -8\right)$

Find the ratio in which the line segment joining $A\left(1, -5\right)$ and $B\left(-4, 5\right)$ is divided by the $x$ axis. Also find the coordinate of the point of division.

If the four vertices of a Rhombus are $\mathrm{P} (1, -3), \mathrm{Q} (6, -3), \mathrm{R} (3, -7)$ and $\mathrm{S} (\mathrm{a}, \mathrm{b})$ taken in order, then find the co-ordinates of $\mathrm{S}$ using an appropriate 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:

From the graph, if Raipur lies in the middle of the straight line joining Hyderabad and Patna, then the coordinates of Raipur 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:

If a line is joined from the position of the ant to the origin, the mid-point of the line will be at

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 in which Bhopal divides the straight line joining Gandhinagar and Ranchi 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 $A\left(-1, 5\right), B\left(3, 2\right), C\left(0, -2\right), D\left(-4, 1\right)$.

Find the mid-point of diagonal $AC$.

Using the section formula, prove that the three points $\left(-4, 6\right)$, $\left(2, 4\right)$ and $\left(8, 2\right)$ are collinear.

If $\mathrm{P}(5,6)$ is the mid-point of the line segment joining $\mathrm{A}(6,5)$ and $\mathrm{B}(4,\mathrm{y})$, then find $\mathrm{y}$.

In the figure, the midpoints of the large quadrilateral are joined to form the small quadrilateral within:

Find the coordinates of the other three vertices of the larger quadrilateral?

If midpoints of sides of a triangle is $(1,2),(0,-1)$ and $(2,-1),$ then find its vertices.

Find the co-ordinates of point which trisects the line joining point $(11,9)$and $(1,2)$.