Section Formula

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

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 $\left(2, 3\right)\text{ and} \left(-2, -3\right)$

If $\mathrm{A}( -2, -1) , \mathrm{B}( p, 0), \mathrm{C}( 4, q)$ and $\mathrm{D}(1, 2)$ are the vertices of a parallelogram, find the values of $p$ and $q$.​

$P$ is a point on the graph of $y=5x+3.$ The coordinates of a point $Q$ are $\left(3, -2\right).$ If $M$ is the mid point of $PQ,$ then $M$ must lie on the line represented by

$R$ is any point on the graph $9x-3y=12$. The coordinates of point $T$ are $(5,-3)$. If $N$ divides $RT$ in the ratio $1:3$ then coordinates of $R$ are _____.

Which one is the trisection formula to find the coordinates of the points which divide the line segment joining $(1, -2)$ and $(-3, 4)$into three equal parts.

Point $R(h, k)$ divides line segment $AB$ between axes in the ratio $1:2$ where $A$ lies on $X-$axis. Find equation of line

The centroid of a triangle is $\left(2, 4\right)$ and circumcentre is $\left(1, 7\right)$ then find the orthocentre

The ratio in which the line segment joining the points $A\left(1,-5\right)$ and $B\left(3,4\right)$  is divided by $x-$axis from $A$, is ___

The point $\left(7,3\right)$ divides the line segment joining the points $\left(4,-3\right)$ and $\left(8,5\right)$ internally in ratio:

Two vertices of a triangle are $(3, 5)$ and $(-4, -5)$. If the centroid of the triangle is $(4, 3)$. Find the third vertex

If the points $A(6,1),B(8,2),C(9,4)\text{ and }D(p,3)$ are the vertices of a parallelogram, taken in order, then the value of $p$ will be -

A segment $AB$ is divided at a point $P$ such that $\frac{PB}{AB}=\frac{3}{7}$,  then the ratio of $AP:PB$ is

The value of $\text{'}C\text{'}$ if$\left(\frac{C}{2}, 14\right)$ is the mid-point of the line joining the points $(–3, 8) \mathrm{and} (–15, 20)$ is

If the line segment joining $(2, 3) \mathrm{and} (–1, 2)$ is divided internally in the ratio $3 : 4$ by the graph of the equation $x + 2y = k$ then the value of $\text{'}k\text{'}$ is

The line segment joining the points $A(2,-2)$ and $B(-7,4)$ is trisected at the points $P$ and $Q$ ($P$ is nearer to $A$). if coordinates of $P$ and $Q$ are $\left(a,0\right)$ and $(-4, b)$ respectively, then the values of $a$ and $b$ are respectively

What is the ratio in which the point $P\left(n, 6\right)$ divides the join of $A\left(-4, 3\right)$ and $B\left(2, 8\right)?$

In what ratio is the line joining the points $A\left(4, 4\right)$ and $B\left(7, 7\right)$ divided by $P\left(-1, -1\right)?$

The coordinates of the point which divides the line joining $\left(1, -2\right)$ and $\left(4, 7\right)$ internally in the ratio $1:2$ are