Q) The class X students school in krishnagar 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 triangular grassy lawn in the plot as shown in the figure. The students are to sow seeds of flowering plants on the remaining area of the plot.

1. Taking A as origin, find the coordinates of P
a) (4,6)                   b) (6,4)                         c) (0,6)                   d) (4,0)

2. What will be the coordinates of R, if C is the origin?
a) (8,6)                  b) (3,10)                        c) (10,3)                d) (0,6)

3. What will be the coordinates of Q, if C is the origin?
a) (6,13)               b) (-6,13)                        c) (-13,6)              d) (13,6)

4. Calculate the area of the triangles if A is the origin
a) 4.5                    b) 6                                c) 8                       d) 6.25

5. Calculate the area of the triangles if C is the origin
a) 8                       b) 5                                 c) 6.25                 d) 4.5

Ans:

1. coordinates of P:

From the diagram, if we take A as origin, we get following:

Coordinates of point P: (4, 6)

Therefore, option (a) is correct.

2. Coordinates of point R if C is origin:

From the diagram, if we take C as origin, we get following:

Coordinates of point R: (10, 3)

Therefore, option (c) is correct.

3. Coordinates of point Q if C is origin:

From the diagram, if we take C as origin, we get following:

Coordinates of point Q: (13, 6)

Therefore, option (d) is correct.

4. Area of the triangle if A is origin:

From the diagram, if we take A as the origin, we get the following coordinates for Δ PQR :

Coordinates of point P: (4, 6)

Coordinates of point Q: (3, 2)

Coordinates of point R: (6, 5)

We know that if a triangle is formed by points (x₁, y₁); (x₂, y₂) and (x₃, y₃), then the area of such an triangle is given by :

A = [x₁ (y₂ – y₃) + x₂ (y₃ – y₁) + x₃ (y₁ – y₂)]

∴  Area of the Δ PQR for above coordinates of P, Q & R:

=  [4 (2 – 5) +3 (5 – 6) + 6 (6 – 2)]

= [- 12 – 3 + 24] = = 4.5 sq m

Therefore, option (a) is correct.

5. Area of the triangle if C is origin:

From the diagram, if we take C as the origin, we get the following coordinates for Δ PQR :

Coordinates of point P: (12, 2)

Coordinates of point Q: (13, 6)

Coordinates of point R: (10, 3)

We know that if a triangle is formed by points (x1, y1); (x2, y2) and (x3, y3), then the area of such an triangle is given by :

[x₁ (y₂ – y₃) + x₂ (y₃ – y₁) + x₃ (y₁ – y₂)]

∴  Area of the Δ PQR for above coordinates of P, Q & R:

= [12 (6 – 3) + 13 (3 – 2) + 10 (2 – 6)]

= [ 36 + 13 – 40] = = – 4.5

Since, area can not be negative, hence only positive value will be taken

∴  Area of the Δ PQR = 4.5 sq. sq m

Therefore, option (d) is correct.

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