The Class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Sapling of Gulmohar are planted on the boundary at a distance of 1m from each other. There is a triangular grassy lawn in the plot as shown in the Fig. The students are to sow 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 ΔPQR if C is the origin?

(i) If we take A as origin then AD is X-axis and AB is the Y-axis. Then the coordinates of the vertices of ΔPQR is P=(4,6), Q=(3,2), R=(6,5)
(ii) If we take C as origin then CB is X-axis and CD is the Y-axis. Then the coordinates of the vertices of ΔPQR is P=(12,2), Q=(13,6), R=(10,3)
Area of the triangle = 1/2[x1(y2−y3) + x2(y3−y1) + x3(y1−y2)]
First case- Area of ΔPQR = 1/2[4(2−5) + 3(5−6) + 6(6−2)] = 1/2(4 × −3 + 3 × −1 + 6 × 4) = 1/2(−12 − 3 + 24) = 9/2 Sq. unit
Second case- Area of ΔPQR = 1/2[12(6−3) + 13(3−2) + 10(2−6)] = 1/2(12 × 3 + 13 × 1 + 10 × −4) = 1/2(36 + 13 − 40) = 9/2 Sq. unit
Hence we observed that the area of the triangle in both cases is equal.