Some Application of Section Formula

If the $\mathrm{x}$-coordinate of a point, which is equidistant from the three vertices $\mathrm{A}(2\mathrm{x}, 0), \mathrm{O}(0, 0)$ and $\mathrm{B}(0, 2\mathrm{y})$ of $△\mathrm{AOB}$ is $\mathrm{y}$, find the sum of the coordinates of point.

If $\mathrm{A}(1,2), \mathrm{B}(4,3)$ and $\mathrm{C}(6,6)$ are the three vertices of a parallelogram $\mathrm{ABCD}$. If the coordinates of $\mathrm{D}$ are $\left(\mathrm{x},\mathrm{y}\right)$, then find the value of $\left(\mathrm{x}+\mathrm{y}\right)$.

If $\mathrm{P}(2,\mathrm{p})$ is the mid-point of the line segment joining the points $\mathrm{A}(6,-5)$ and $\mathrm{B}(-2,11)$, then find the value of $\mathrm{p}$.

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

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 $\mathrm{y}$.

Write the ratio in which the line segment joining the points $\mathrm{A}(3,-6)$, and $\mathrm{B}(5,3)$ is divided by $\mathrm{x}$-axis. If the sum of the terms in the ratio is $k,$ find $k ?$

Vertices of a triangle have co-ordinates $(-8,7), (\mathrm{x},\mathrm{y})$ and$(9,4)$. If the centroid of the triangle is at the origin, then find $\mathrm{m}$ such that $\mathrm{xy}=\mathrm{m}$.

If the mid-point of the segment joining $\mathrm{A}(\mathrm{x},\mathrm{y}+1)$ and $\mathrm{B}(\mathrm{x}+1,\mathrm{y}+2)$ is $\mathrm{C}\left(\frac{3}{2},\frac{5}{2}\right)$ and $\mathrm{x}+\mathrm{y}=\mathrm{m}$, then find $\mathrm{m}.$

If the centroid of the triangle formed by points $\mathrm{P}(\mathrm{a},\mathrm{b}), \mathrm{Q}(\mathrm{b},\mathrm{c})$ and $\mathrm{R}(\mathrm{c},\mathrm{a})$ is at the origin, then find value of  $\frac{{\mathrm{a}}^2}{\mathrm{bc}}+\frac{{\mathrm{b}}^2}{\mathrm{ca}}+\frac{{\mathrm{c}}^2}{\mathrm{ab}}$?

If the centroid of the triangle formed by points $\mathrm{P}(\mathrm{a},\mathrm{b}), \mathrm{Q}(\mathrm{b},\mathrm{c})$ and $\mathrm{R}(\mathrm{c},\mathrm{a})$ is at the origin, what is the value of $\mathrm{a}+\mathrm{b}+\mathrm{c}$?

If the coordinates of the point dividing line segment joining points $(2,3)$ and $(3,4)$ internally in the ratio $1:5$ is $\left(\frac{a}{6},\frac{b}{6}\right)$, then find $b-a$.

If $A(-1,3), B(1,-1)$ and $C(5,1)$ are the vertices of a triangle $△ABC$, then the length of the median through vertex $A$ is $k$ units. Find the value of $k$.

Write the ratio $\left(\mathrm{m}:\mathrm{n}\right)$ in which the line segment joining points $\left(2,3\right)$ and $\left(3,-2\right)$ is divided by $\mathrm{x}$-axis. If  $\mathrm{m}+\mathrm{n}=\mathrm{p}$ then find $\mathrm{p}$.

Prove that the lines joining the middle points of the opposite sides of a quadrilateral and the join of the middle points of its diagonals meet in a point and bisect one another.

Prove analytically that the line segment joining the middle points of two sides of a triangle is equal to half of the third side.

If $(-2,3), (4,-3) \mathrm{and} (4,5)$ are the mid-points of the sides of a triangle, then find the coordinates of its centroid.

$A(3,2) \mathrm{and} B(-2,1)$are two vertices of a triangle $ABC$ whose centroid $G$ has the coordinates $\left(\frac{5}{3},\frac{-1}{3}\right)$. Find the coordinates of the third vertex $C$ of the triangle.

Find the third vertex of a triangle, if two of its vertices are at $(-3,1) \mathrm{and} (0,-2)$ and the centroid is at the origin.

Two vertices of a triangle are $(1,2), (3,5)$ and its centroid is at the origin. Find the coordinates of the third vertex.

Find the centroid of the triangle whose vertices are $(-2,3), (2,-1), (4,0)$.