The length of a rectangle is twice its breadth. If its length is decreased half of the 10 cm and the breadth is increased by half of the 10 cm cm, the area of the rectangle is increased by 5 sq cm, more than 70 sq cm. Find the length of the rectangle.

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

Ratio of similar triangle
= Ratio of the square of corresponding sides
= (3x)² / (4x)² = 9x² / 16x²
= 9/16 = 9 : 16

Area of square = (Side)² = 20²
= 400 sq cm

? Area of rectangle
= 1.8 x 400 = 720 sq cm

Let length and breadth of rectangle be 5k and k respectively.
Then, according to the question,
5k x k = 720
? 5k² = 720
? k² = 720/5 = 144
? k = ?144 = 12 cm

Perimeter of rectangle = 2(5k + k) = 12k
= 12 x12 = 144 cm

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