Question

Question 5
The class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted on the boundary at a distance of 1 m from each other. There is a triangular lawn in the plot as shown in the figure. The students are to sow the seeds of flowering plants on the remaining area of the plot.
(a) Taking A as origin, find the coordinates of the vertices of the triangle.
(b) What will be the coordinates of the vertices of triangle PQR if C is the origin.
(c) Also calculate the areas of the triangles in these cases. What do you observe?

Answer 1

(a) If A is the origin, then the cordinates are: P = (4, 6), Q = (3, 2), R (6, 5) respectively.
(b) The coordinates of vertices of the triangle PQR, if C is the origin: P = (-12, -2), Q = (-13, -6), R = (-10, -3)
(c) Area of triangle PQR in case of origin A:
$=\frac{1}{2}\left[x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right]$
$=\frac{1}{2}\left[4\right(2-5)+3(5-6)+6(6-2\left)\right]$
$=\frac{1}{2}(-12-3+24)$
$=\frac{9}{2}squnit$
Area of triangle PQR in case of origin C:
$=\frac{1}{2}\left[x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right]$
$=\frac{1}{2}[-12(-6+3)+-13(-3+2)+(-10\left)\right(-2+6\left)\right]$
$=\frac{1}{2}(36+13-40)$
$=\frac{9}{2}squnit$
Area is same in both the cases because triangle remains the same no matter which point is considered as origin.