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.
(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)]
= 1/2 [4(2 – 5) + 3 (5 – 6) + 6 (6 – 2)]
= 1/2 (- 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)]
= 1/2 [ 12(6 – 3) + 13 ( 3 – 2) + 10( 2 – 6)]
= 1/2 ( 36 + 13 – 40)
= 9/2 sq unit
This implies, Area of triangle PQR at origin A = Area of triangle PQR at origin C