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 grassy lawn in the plot as shown in the following figure. 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 Δ PQR if C is the origin? Also calculate the areas of the triangles in these cases. What do you observe?
Class 10 - Math - Coordinate Geometry Page 171
Answers
Area of a Δ ABC when its three vertices are given.
If A(x1, y1), B(x2, y2) and C(x3, y3) are the vertices of a triangle then its area is given by
Area of ΔABC = 1 /2 [x1(y2 - y3) + x2( y3 - y1)+ x3(y1 - y2)] square units.
If the area of triangle is zero then the points are called collinear points.
If three points A(x1, y1),B(x2, y2) and C(x3, y3) are collinear then
Area of triangle =0, [x1(y2 - y3) + x2( y3- y1)+ x3(y1 - y2)] = 0.
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Solution:
i) when A is taken as origin , AD & AB as coordinates axes i.e X & Y axes. Coordinates of P , Q & R are (4,6) , (3,2), (6,5)
ii) i) when C is taken as origin & CB & CD as coordinates axes i.e X & Y axes. Coordinates of P , Q & R are (12,2) , (13,6), (10,3)
Area of Δ according to both cases are as follows
Case I:
Area of ΔPQR in case of origin A & AD & AB as coordinate axes.
= ½ [x1(y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)]
= ½ [4(2 – 5) + 3 (5 – 6) + 6 (6 – 2)]
= ½ ( - 12 – 3 + 24
Area of ΔPQR= 9/2 sq unit
Case II
Area of Δ PQR in case of origin C & CB & CD as coordinate axes.
= ½ [x1(y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)]
= ½ [ -12( - 6 + 3) + - 13 (- 3 + 2) + - 10( - 2 + 6)]
= ½ ( 36 + 13 – 40)
Area of Δ PQR= 9/2 sq unit
Hence, we observe that area of Δ is same in both case because triangle remains the same no matter which point is considered as origin.----------------------------------------------------------------------------------------------------Hope this will help you...---------------------------------------------------------------------------------------------------
Solution:
Hole it helps..u....
