S L Loney Solutions for Chapter: Coordinates, Lengths of Straight Lines and Areas of Triangles, Exercise 1: EXAMPLES I

The line joining the points $(1,-2)$ and $\left(-\mathrm{3,}4\right)$ is trisected; find the co-ordinates of the points of trisection.

The line joining the points $\left(-6, 8\right)$ and $\left(8, -6\right)$ is divided into four equal parts; find the co-ordinates of the points of section.

The co-ordinates of the vertices of a triangle are $\left(x_1, y_1\right), \left(x_2, y_2\right)$ and $\left(x_3, y_3\right).$ The line joining the first two is divided in the ratio $l:k$ and the line joining the points of division to the opposite angular point is then divided in the ratio $m:k+l$. Find the co-ordinates of the latter point of section.

Prove that co-ordinates $x$ and $y$ of the midpoint of the line joining the point $\left(\mathrm{2,} 3\right)$ to the point $\left(\mathrm{3,} 4\right)$ satisfy the equation $x-y+1=0$.

If $G$ is the centroid of a triangle $ABC$ and $O$ is any other point, prove that
$3\left(G{A}^2+G{B}^2+G{C}^2\right)=B{C}^2+C{A}^2+A{B}^2$
and $O{A}^2+O{B}^2+O{C}^2=G{A}^2+G{B}^2+G{C}^2+3G{O}^2$.

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

$A, B, C, D...$  are $n$ points in a plane whose coordinates are $\left(x_1, y_1\right), \left(x_2, y_2\right), \left(x_3, y_3\right),...$ $AB$ is bisected at the point $G_1; G_{1}C$ is divided at $G_2$ in the ratio $1:2$; $G_{2}D$ is divided at $G_3$ in the ratio $1:3;$ $G_{3}E$ at $G_4$ in the ratio $1:4$, and so on until all the points are exhausted. Show that the coordinates of the final point so obtained are,

$\frac{x_1+x_2+x_3+....+x_n}{n}$ and $\frac{y_1+y_2+y_3+....+y_n}{n}$.

(This point is called the centre of the mean position of the $n$ given points.)

Prove that a point can be found which is at the same distance from each of the four points.

$\left(a{m}_1, \frac{a}{m_1}\right)$, $\left(a{m}_2, \frac{a}{m_2}\right)$, $\left(a{m}_3, \frac{a}{m_3}\right)$ and $\left(\frac{a}{m_1{m}_2{m}_3}, a{m}_1{m}_2{m}_3\right)$.