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The diagonals of a field in the form of a quadrilateral are $106 m$ and $80 m$ and intersect each other at right angles. Find the cost of cultivating the field at the rate of Rs. $25.50$ per $100 m 2$

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R s . 1081.20

R s . 981.20

R s . 1181.20

R s .1020.34

answer is A.

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Detailed Solution

Take ABCD as the example quadrilateral.
Question Image

A quadrilateral's diagonals are the line segments that connect its two opposite vertices; therefore, AC and BD are its diagonals. At O, the diagonals come together.
The field's diagonals, which are claimed to be

106 m

and

80 m

in length, cross at right angles.
Consider,

A C = 106 m

and

B D = 80 m

Let

O A = a O B = b O C = c O D = d

The area of a right triangle is equal to half the product of the base and the height because triangle

O A B, O B C, O C D, O D A

is a right-angled triangle.
Or

A = 12 × b × h

So,

ar (Δ O A B) = 12 × O A × O B ⇒ a r (Δ O A B) = 12 × a × b = a b 2 … (i)

Similarly,

ar (Δ O B C) = 12 × O C × O B ⇒ a r (Δ O B C) = 12 × c × b = c b 2 … (ii) ar (Δ O C D) = 12 × O C × O D ⇒ a r (Δ O C D) = 12 × c × d = c d 2 … (iii) ar (Δ O A D) = 12 × O A × O D ⇒ a r (Δ O A D) = 12 × a × d = a d 2 … (iv)

Now, the four triangles' total areas add up to the area of the quadrilateral.
Thus,

a r (A B C D) = a r (Δ O A B) + a r (Δ O B C) + a r (Δ O C D) + a r (Δ O A D)

By using above equations, we get

⇒ ar (A B C D) = a b 2 + b c 2 + c d 2 + a d 2 ⇒ ar (A B C D) = a b + b c + c d + a d 2 ⇒ a r (A B C D) = b (a + c) + d (c + a) 2 ⇒ ar (A B C D) = (b + d) (a + c) 2

Now, we will substitute back the values of

a, b, c, d

in the above equation,

⇒ a r (A B C D) = (O B + O D) (O A + O C) 2 ⇒ a r (A B C D) = (B D) (A C) 2

Now, we have the area of a quadrilateral in terms of its diagonals.
Given that,

A C = 106 m

and

B D = 80 m

⇒ a r (A B C D) = (80) (106) 2 m 2 ⇒ a r (A B C D) = 4240 m 2

Thus, the quadrilateral area is

4240 m 2

The price of growing a field is:

R s .25.50

per

100 m 2

So, cost of cultivating

100 m 2

is:

R s .25.50

⇒

Cost of cultivating

1 m 2

is: Rs.

25.50 100

⇒

Cost of cultivating

4240 m 2

is: Rs.

25.50 100 × 4240 =

Rs.

1081.20

Therefore,

R s . 1081.20

is needed to cultivate the field.
So, the option A's value of

R s . 1081.20

is correct.
Hence, Option (1) is the correct Ans:,

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