Given perimeter of the rhombus = 180 cm
∴ One side of the rhombus = \(\frac { 180 }{ 4 } \) = 45 cm
Given area of the rhombus = 315 cm²
b x h = 315
45 x h = 315
= \(\frac { 315 }{ 45 } \)
h = 7cm
Altitude of the rhombus = 7 cm

Given, Area of the Trapezium = 828 cm²
\(\cfrac { 1 }{ 2 } \times h(a+b)=828\)
\(\cfrac { 1 }{ 2 } \times h(19.6+16.4)=828\)
\(\cfrac { 1 }{ 2 } \times h(36)=828\)
h (18) = 828
\(h=\cfrac { 828 }{ 18 } \)
h - = 46 m
Therefore, distance between the parallel sides = 46 cm

Given, height (h) = 14 cm
parallel sides are (a) = 18 cm and (b) = 9 cm
Area of the trapezium = \(\cfrac { 1 }{ 2 } \times h(a+b)\)
= \(\cfrac { 1 }{ 2 } \times 14\left( 18+9 \right) \)
= 7(27)
= 189 sq. cm
Therefore, area of the trapezium is 189 sq. cm.

Let the height of the parallelogram = h cm
Then the base of the parallelogram = 3h cm
Area of the parallelogram = 192 sq. cm
b x h = 192
3h x h = 192
3h² = 192
h² = 64
h x h = 8 x 8
h = 8 cm
base = 3h = 3 x 8 = 24 cm
Therefore, base of the parallelogram is 24 cm and height is 8 cm.

Given: b = 12 m, h = 8 m
Area of the parallelogram = b x h sq.units
= 12 x 8 = 96 sq.m
Therefore, Area of the parallelogram = 96 sq.m.

Given perimeter = 150 cm
Let one side of the parallelogram be 'b' cm
Then the other side = b + 25 cm
b + (b + 25) + b + (b + 25) = 150
b + b + 25 + b + b + 25 = 150
4b + 50 = 150 = 4b =100

∴ One side b = 25 cm
Other side b + 25 = 50 cm

base = 4 times of the height
b = 4h
Area of parallelogram = 576
b x h = 576
4h x h = 576
4h² = 576
h² = I44 cm
h = 12 cm
Ans : Height of the parallelogram is 12 cm.

Given : One of the diagonals of a rhombus is thrice the other
\(\Rightarrow \ \mathrm{d}_{2}=3 \mathrm{~d}_{1}\)       ............(1)
Sum of the length of the diagonals = 24 cm
d₁+ d₂ = 24cm
3 d₁ + 3d₂ = 24
4d₁ = 24
\(d_{1}=\frac{24}{4}=6 \mathrm{~cm}\)
d₁ = 6 cm
d₂  = 3d₁
= 3 x  6
d₂ = 18 cm
Area of the rhombus \(=\frac{1}{2} \mathrm{~d}_{1} \times \mathrm{d}_{2} \text { sq.units } \)
\(=\frac{1}{2} \times 6 \times 18 \)
= 54 sq.cm

In diagram (i)
Consider DF as height (h) and AB as base (b)
Area of the parallelogram = 1470 sq.cm
b x h = 1470
49 x h = 1470
h = 30 
height, DF = 30 cm

In diagram (ii),
Consider BE as height (h) and AD as base (b) 
Area of the parallelogram = 1470 sq.cm
b x h = 1470
35 x h = 1470
height BE = 42 cm
Ans: DF = 30 cm and BE = 42 cm

Length of one diagonal \(=\frac{1}{2}\) length of other diagonal
\(\mathrm{d}_{2}=\frac{1}{2} \mathrm{~d}_{1}\)
Area of the rhombus = 576
\(\frac{1}{2} \times\left(d_{1} \times d_{2}\right)=576 \)
\(\frac{1}{2} \times d_{1} \times \frac{1}{2} d_{1}=576 \)
d₁ x d₁ = 576 x 2 x 2
d₁ x d₁ = 24 x 24 x 2 x 2
d₁ = 24 x 2
d₁ = 48 cm
\(\mathrm{d}_{2}=\frac{1}{2} \mathrm{~d}_{1}=\frac{1}{2} \times 48\)
d₂ = 24 cm
The length of the diagonals are 24 cm and 48 cm.

Given : Area of a parallelogram = Area of a square
b x h = a x a
b x 18 = 48 x 48
\(b=\frac{48 \times 48}{18}\)
Base (b) = 128 m

Given : A = 1080 sq.cm, a = 55.6 cm, b = 34.4 cm
Distance between them, h = ?
Area of trapezium = 1080 sq.cm
\(\frac{1}{2} \times h(a+b) =1080 \)
\(\frac{1}{2} \times h(55.6+34.4) =1080 \)
\(\frac{1}{2} \times h \times 90 =1080 \)
\(h =\frac{1080 \times 2}{90} \)
Distance, h = 24 cm.

Given : Area = 100 sq.cm, d₁ = 8 cm, d₂ = ?
Area of rhombus = 100 sq.cm
\(\frac{1}{2} \times\left(d_{1} \times d_{2}\right)=100 \)
\(\frac{1}{2} \times\left(8 \times d_{2}\right)=100 \)
8 x d₂ = 100 x 2
\(\mathrm{d}_{2}=\frac{100 \times 2}{8} \Rightarrow \mathrm{d}_{2}=25 \mathrm{~cm}\)

h = 12 m,b = 18 m
Area of a parallelogram (fabric) = b x h = 18 x 12
= 216 m² (or) sq.m
She cuts the fabric into 4 equal parallelograms
Area of a new parallelogram = 54 m²

(i) Area of a parallelogram
= b x h sq.units
= 11 x 3 = 33 cm²
Perimeter of the parallelogram
= sum of the length of the four sides.
= 11+ 4 + 11 + 4 = 30 cm.
(ii) Area of a parallelogram
= b x h sq.units
= 7 x 10 = 70 cm²
Perimeter of the parallelogram
= sum of the length of the four sides.
= 13 + 7 + 13 + 7 = 40 cm

Given, the length of the diagonals of a rhombus shaped tile are 40 cm and 25 cm
The area of one tile = \(\cfrac { 1 }{ 2 } \times \left( { d }_{ 1 }\times { d }_{ 2 } \right) sq.units\)
= \(\cfrac { 1 }{ 2 } \times 40\times 25\)
= 500 sq.cm
Therefore, the area of 200 such tiles = 200 x 500
= 100000 sq. cm
= \(\cfrac { 100000 }{ 10000 } \)(1sq.m = 1000 sq.cm)
= 10 sq.m
Therefore, the cost of polishing 200 such tiles at the rate of Rs 45 per sq. cm
= 10 x 45 = Rs. 450.

Base = 17 cm, height = 8 cm
Area of the rhombus = b x h sq. units
= 17 x 8 = 136
Therefore, area of the rhombus = 136 sq. cm