91Ó°ÊÓ

Find the area of the quadrilateral whose vertices, taken in order, are \((-4,-2),(-3,-5)\), \((3,-2)\) and \((2,3)\)

Check whether \((5,-2),(6,4)\) and \((7,-2)\) are the vertices of an isosceles triangle.

Find the ratio in which the line segment joining \(\mathrm{A}(1,-5)\) and \(\mathrm{B}(-4,5)\) is divided by the \(x\) -axis. Also find the coordinates of the point of division.

Let \(\mathrm{A}(4,2), \mathrm{B}(6,5)\) and \(\mathrm{C}(1,4)\) be the vertices of \(\Delta \mathrm{ABC}\). (i) The median from A meets \(\mathrm{BC}\) at \(\mathrm{D}\). Find the coordinates of the point \(\mathrm{D}\). (ii) Find the coordinates of the point \(\mathrm{P}\) on \(\mathrm{AD}\) such that \(\mathrm{AP}: \mathrm{PD}=2: 1\) (iii) Find the coordinates of points \(Q\) and \(R\) on medians \(B E\) and \(C F\) respectively such that \(\mathrm{BQ}: \mathrm{QE}=2: 1\) and \(\mathrm{CR}: \mathrm{RF}=2: 1\). (iv) What do yo observe? [Note : The point which is common to all the three medians is called the centroid and this point divides each median in the ratio \(2: 1 .]\) (v) If \(\mathrm{A}\left(x_{1}, y_{1}\right), \mathrm{B}\left(x_{2}, y_{2}\right)\) and \(\mathrm{C}\left(x_{3}, y_{3}\right)\) are the vertices of \(\Delta \mathrm{ABC}\), find the coordinates of the centroid of the triangle.

If \(A\) and \(B\) are \((-2,-2)\) and \((2,-4)\), respectively, find the coordinates of \(P\) such that \(\mathrm{AP}=\frac{3}{7} \mathrm{AB}\) and \(\mathrm{P}\) lies on the line segment \(\mathrm{AB}\).

Find the area of a rhombus if its vertices are \((3,0),(4,5),(-1,4)\) and \((-2,-1)\) taken in order. [Hint : Area of a rhombus \(=\frac{1}{2}\) (product of its diagonals)]