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. 7.14. 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 ∆ PQR if C is the origin? Also calculate the areas of the triangles in these cases. What do you observe?
Solution::
(i) Taking A as origin, we will take AD as the x-axis and AB as the y-axis.
It can be observed from the figure that the coordinates of point P, Q, and R are (4, 6), (3, 2), and (6, 5) respectively.
Let P(x₁, y₁) = (4, 6), Q(x₂, y₂) = (3 , 2) and R (x₃, y₃ ) = (6, 5)
Area of a triangle = 1/2 [x₁ (y₂ - y₃) + x₂ (y₃- y₁) + x₃ (y₁ - y₂)] .....(1)
By substituting the values of vertices P, Q, R in the Equation (1),
Area of triangle PQR = 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 square units
(ii) Taking C as origin, CB as x-axis, and CD as the y-axis
The coordinates of vertices P, Q, and R are (-12, -2), (-13, -6), and (-10, -3) respectively.
Let P(x₁, y₁) = (-12, -2) , Q(x₂, y₂) = (-13, -6) and R(x₃, y₃) = (-10, -3)
Area of a triangle = 1/2 [x₁ (y₂ - y₃) + x₂ (y₃ - y₁) + x₃ (y₁ - y₂)] .....(1)
By substituting the values of vertices P, Q, R in the Equation (1),
Area of triangle PQR = 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 square units
It can be observed that the area of the triangle is the same in both cases.
☛ Check: NCERT Solutions for Class 10 Maths Chapter 7
Video Solution:
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. 7.14. 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 ∆ PQR if C is the origin? Also calculate the areas of the triangles in these cases. What do you observe?
Maths NCERT Solutions Class 10 Chapter 7 Exercise 7.4 Question 5
Summary:
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 above figure. The students are to sow seeds of flowering plants on the remaining area of the plot. (i) aking A as origin, the coordinates of the vertices are P, Q, and R are (4, 6), (3, 2), and (6, 5) respectively and (ii) The coordinates of the vertices of ∆PQR, if C is the origin, are P, Q, and R are (-12, -2), (-13, -6), and (-10, -3). Also, the areas of the triangles in both cases are 9/2 square units.
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