The area of rectangle is 1.8 times the area of a square. The length of the rectangle is 5 times the breadth. The side of the square is 20 cm. What is the perimeter of the rectangle?

Given that, l = 2b [Here l = length and b = breadth]

Decrease in length = Half of the 10 cm = 10/2 = 5 cm
Increase in breadth = Half of the 10 cm = 10/2 = 5 cm
Increase in the area = (70 + 5) = 75 sq cm

According to the question,
(l - 5) (b + 5) = lb + 75
? (2b - 5) (b + 5) = 2b² + 75 [since l = 2b]
? 5b - 25 = 75
? 5b = 100
? b = 100/ 5 = 20
? l = 2b = 2 x 20 = 40 cm

We know that, the radius of a circle inscribed in a equilateral triangle = a/[2?3]
Where, a be the length of the side of an equilateral triangle.

Given that, area of a circle inscribed in an equilateral tringle = 154 cm²
? ?(a/2?3)² = 154
? (a/2?3)² = 154 x (7/22) = (7)a²
? a = 42?3 cm

Perimeter of an equilateral triangle = 3a
= 3(14?3)
= 42?3 cm

Area of equilateral triangle = ?3a²/4 = x ......(i)

And perimeter = 3a = y
? a = y/3 ....(ii)

Now, Putting the value of a from Eq. (ii) in Eq. (i). we get
?3 (y/3)²/4 = x
? x = ?3 x y²/36
? x = y²/3?3x = y²/12?3
12?3 x = y²
On squaring both sides, we get
y⁴ = 432x²

Let the radius of circle is 'r' and a side of a square is 'a',
then given condition
2?r = 4a
? a = ?r/2

? Area of square = (?r/2)² = ?² /4r² = 9.86r²/4 = 2.46r²
and area of circle = ?r² = 3.14;r²
and let the side of equilateral triangle is x.
Then, given condition,
3x = 2?r
? x = 2?r/3

? Area of equilateral triangle = ?3/4 x ²
= ?3/4 x 4?²r²/9
= ?²/3?3r²
= 1.89r²

Hence, Area of circle > Area of square > Area of equilateral triangle.

Let there be n sides of the polygon. Then it has n vertices.
The total number of straight lines obtained by joining n vertices by talking 2 at a time is ⁿC₂
These ⁿC₂ lines also include n sides of polygon.
Therefore, the number of diagonals formed is ⁿC₂ - n.
Thus, ⁿC₂ - n = 44
? [n(n - 1)/2] - n = 44
? ( n² - 3n) / 2 = 44
? n² - 3n = 88
? n² - 3n - 88 = 0
?(n - 11) (n + 8) = 0
? n = 11

Area of square plate = (Side)²
= (2d)²
= 4d²

Area of circular plate = ? (d/2)²
= ?d²/4

? Number of square plates
= [(4d²)/4] / [(?d²)/4]
= (4 x 4)/?
? 5
Since, nearest integer value is 5.

Let one diagonal be k.
Then, other diagonal = (60k/100) = 3k/5 cm
Area of rhombus =(1/2) x k x (3k/5) = (3/10)
= 3/10 (square of longer diagonal)
Hence, area of rhombus is 3/10 times.

? a³ = 512 = 8 x 8 x 8
? a = 8 cm
? Surface area = 6a²
=[6 x (8)²] cm²
=384 cm²

Surface area = 6a² = 726
? a² = 121
? a = 11 cm
? Volume of the cube = (11 x 11 x 11) cm³
= 1331 cm³

Let the edge of original cube = x cm
Edge of new cube = (2x) cm
Ratio of their volumes = x³ : (2x)³
= x³ : 8x³
= 1 :8

Thus the volumes be comes 8 times.