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.

Area of triangle PQR in case of origin A:

Area of a triangle = \( \frac{1}{2} \ | \ [4(2 – 5) + 3 (5 – 6) + 6 (6 – 2)] \ | \ \)
\( = \ \frac{1}{2} \ | \ [– 12 – 3 + 24] \ | \ \)
\( = \ \frac{9}{2} \) sq. units

(ii) Taking C as origin, coordinates of vertices P, Q and R are,

From figure:
P = (12, 2)
Q = (13, 6)
R = (10, 3)

Here CB is the x-axis and CD is the y-axis.

Area of triangle PQR in case of origin C:

Area of a triangle = \( \frac{1}{2} \ | \ [ 12(6 – 3) + 13 ( 3 – 2) + 10( 2 – 6)] \ | \ \)
\( = \frac{1}{2} \ | \ [36 + 13 – 40] \ | \ \)
\( = \ \frac{9}{2} \) 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.