Question

The diagonals of a rectangle enclose an angle of measure 78°. If the perimeter of the
rectangle is 36 cm, what is its area?​

Answer 1

Given :  diagonals of the rectangle is enclose an angle of measure 78°

the perimeter of the rectangle is 36cm

To Find :  Area of rectangle

Solution:

Let say one side of rectangle = a

then other side  =  (36 - 2a)/2  = 18 - a   cm

diagonals of the rectangle enclose 78°

hence angle Diagonal has with base = (180° - 78°)/2

= 51°

Tan 51°  =  a/(18 - a)

=> 1.235 = a/(18 - a)

=> 22.23 - 1.235a = a

=> 22.23 = 2.235a

=> a = 9.946

=> 18 - a =  8.054

Area = 9.946 x 8.054 = 80.1  cm²

or Tan 51°  =   (18 - a)/a

=>  1.235a = 18 - a

=> 2.235a = 18

=> a = 8.054

18-a = 9.946

Area = 9.946 x 8.054 = 80.1 cm²

Another method

Area = (1/2)d²sin78°

d = diagonal   d² = a² + b²   = (a + b)²  - 2ab     = 18²  - 2A

A = area of reactngel

=> A  = (1/2) ( 18²  - 2A) sin78°

=> 2A =  18² sin78° - 2A sin78°

=> 2A(1 +  sin78°) =  18² sin78°

=> A =  18² sin78°/ 2(1 +  sin78°)

=> A = 80.1 cm²

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Answer 2

Answer:

Area of rectangle is 80.1 cm².

Step-by-step explanation:

Given: Diagonal of rectangle encloses $78^\circ$ and perimeter=36 cm.

To find: Area of rectangle.

We know that area of rectangle=1/2$d^{2}sin\theta$, $\theta$ is angle between diagonal.

Let one side of rectangle=x cm.

Then, using perimeter of rectangle=2(l+b)=36

We have other side= 18-x

diagonal d² = a² + b² = $x^{2} +(18-x)^{2}$= 18²-2A

Where, A = area of rectangle

Area = (1/2)d²sin78°

A  = (1/2) ( 18²  - 2A) sin78°

2A =  18² sin78° - 2A sin78°

2A(1 +  sin78°) =  18² sin78°

A =  18² sin78°/ 2(1 +  sin78°)

A = 80.1 cm²

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