. 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 fig. 7.14. The students are to sow the seeds of flowering plants on the remaining area of the plot. (i) Taking A as origin, find the coordinates of the vertices of the triangle. (ii) What will be the coordinates of the vertices of triangle PQR if C is the origin? Also calculate the areas of the triangles in these cases. What do you observe?

(i) Taking A as origin, coordinates of the vertices P, Q and R are, From figure: P = (4, 6), Q = (3, 2), R (6, 5) Here AD is the x-axis and AB is the y-axis. (ii) Taking C as origin, Coordinates of vertices P, Q and R are ( 12, 2), (13, 6) and (10, 3) respectively. Here CB is the x-axis and CD is the y-axis. Find the area of triangles: Area of triangle PQR in case of origin A: Using formula: Area of a triangle = 1/2 × [x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)] = ½ [4(2 – 5) + 3 (5 – 6) + 6 (6 – 2)] = ½ (- 12 – 3 + 24 ) = 9/2 sq unit (ii) Area of triangle PQR in case of origin C: Area of a triangle = 1/2 × [x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)] = ½ [ 12(6 – 3) + 13 ( 3 – 2) + 10( 2 – 6)] = ½ ( 36 + 13 – 40) = 9/2 sq unit This implies, Area of triangle PQR at origin A = Area of triangle PQR at origin C Area is same in both case because triangle remains the same no matter which point is considered as origin.