S L Loney Solutions for Chapter: Coordinates, Lengths of Straight Lines and Areas of Triangles, Exercise 2: EXAMPLES II

Find the area of the triangle the co-ordinates of whose angular points are

$\left(a, c+a\right), \left(a, c\right) \text{and }\left(-a, c-a\right)$.

Find the area of the triangle the co-ordinates of whose angular points are $\left(a{m}_1, \frac{a}{m_1}\right), \left(a{m}_2,\frac{a}{m_2}\right)$ and $\left(a{m}_3,\frac{a}{m_3}\right)$.

By showing that the area of the triangle formed by them is zero, prove that the following sets of three points are in a straight line: $\left(\mathrm{1,} 4\right), \left(3, -2\right)$ and $\left(-\mathrm{3,} 16\right)$.

By showing that the area of the triangle formed by them is zero, prove that the following sets of three points are in a straight line: $\left(-\frac{1}{2}, 3\right), \left(-\mathrm{5,} 6\right)$ and $\left(-\mathrm{8,} 8\right)$.

By showing that the area of the triangle formed by them is zero, prove that the following points are in a straight line: $\left(a, b+c\right), \left(b, c+a\right)$ and $\left(c, a+b\right)$.

Find the area of the quadrilateral the co-ordinates of whose angular points, taken in order, are $\left(\mathrm{1,} 1\right), \left(\mathrm{3,} 4\right), \left(5, -2\right)$ and $\left(4, -7\right)$.

Find the area of the quadrilateral the co-ordinates of whose angular points, taken in order, are $\left(-\mathrm{1,} 6\right), \left(-3, -9\right), \left(5, -8\right)$ and $\left(\mathrm{3,} 9\right).$

If $O$ is the origin, and if the co-ordinates of any two points $P_1$ and $P_2$ are respectively $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right),$ prove that
$O{P}_1·O{P}_2·\cos \angle \left(P_{1}O{P}_2\right)=x_1{x}_2+y_{\mathit{1}}{y}_2.$