For \(\Delta \)ABC
 a = 4 cm, b = 5 cm, c = 3 cm
\(\because \) a² + c² = b²
 
\(\therefore \) \(\Delta \)ABC is right angled with \(\angle B=90°\).
\(\therefore \)  Area of right triangle ABC = \(\frac { 1 }{ 2 } \times \)Base \(\times \) Height
\(=\frac { 1 }{ 2 } \times 3\times 4=6\) cm²
For \(\Delta \)ACD     
a = 4 cm, b = 5 cm, c = 5 cm
 \(\therefore s=\frac { a+b+c }{ 2 } =\frac { 4+5+5 }{ 2 } =\frac { 14 }{ 2 } =7\) cm
 \(\therefore \) Area of the ACD \(=\sqrt { s(s-a)(s-b)(s-c) } \)
 \(=\sqrt { 7(7-4)(7-5)(7-5) }\)
\( \\ =\sqrt { 7(3)(2)(22) }\)
\(=\ 2\sqrt { 21 } \) cm²
= 2\(\times \) 4.6 cm² (approx.) =9.2 cm² (approx.)
\(\therefore \) Area of the equilateral ABCD = Area of \(\Delta \) ABC+Area of \(\Delta \)ACD
= 6 cm² + 9.2 cm² = 15.2 cm² (approx.)

For \(\Delta \)ABC   a = 30 m, b = 48 m, c = 30 m
\(\therefore s=\frac { a+b+c }{ 2 } =\frac { 30+48+30 }{ 2 } =\frac { 108 }{ 2 } =54\) m
 
\(\therefore \)  Area of \(\Delta \)ABC = \(\sqrt { s(s-a)(s-b)(s-c) } \)
\(=\sqrt { 54(54-30)(54-48)(54-30) } \)
\(\\ =\sqrt { 54(24)(6)(24) } \)
\(\\ =\sqrt { \left( 9\times 6 \right) (24)\left( 6 \right) \left( 24 \right) } \)
 \(=3\times 6\times 24=432\) m²
\(\therefore \)  Area of the rhombus = 2 area of \(\Delta \)ABC
\(=2\times 432=864\) m^2.
Area of grass for 18 crows = 864 m².
Area of grass for 1 cow = \(\frac { 864 }{ 18 } \) m² = 48 m².

Let the given field be in the shape of a trapezium ABCD in which AB=25 m, CD= 10 m, BC = 13 m and AD = 14 m.
From D, draw DE 11 BC meeting AB at E. Also, draw DF \(\bot \) AB.
\(\therefore \) DE = BC =13 m
AE = AB - EB = AB - DC
= 25 - 10 = 15 m

For AED
a = 14 m, b = 13 m, c=  15 m
\(\therefore s=\frac { a+b+c }{ 2 } =\frac { 14+13+15 }{ 2 } =\frac { 42 }{ 2 } =21\) m
\(\therefore \)    Area of the \(\Delta \)AED  \(=\sqrt { s(s-a)(s-b)(s-c) } \)
 \(=\sqrt { 21(21-14)(21-13)(21-15) }\)
\( \\ =\sqrt { 21(7)(8)(6) } =\sqrt { \left( 7\times 3 \right) (7)\left( 4\times 2 \right) \left( 2\times 3 \right) } \)
\(=7\times 3\times 2\times 2=84\) m²
\(\Rightarrow \frac { 1 }{ 2 } \times \)AE\(\times \)DE = 84
\(\Rightarrow \ \frac { 1 }{ 2 } \times \)15\(\times \)DF = 84
\(\Rightarrow \) DF = \(\frac { 84\times 2 }{ 15 } \)
\(\Rightarrow \) DF=  \(\frac { 56 }{ 5 } \) m = 11.2 m
\(\Rightarrow\) Height of the trapezium is 11.2 m.
\(\therefore \) Area of parallelogram EBCD = Base \(\times \) Height
= EB\(\times \) DF = 10 \(\times \)\(\frac { 56 }{ 5 } \) = 112 m²
\(\therefore \) Area of the field = Area of AED + Area of parallelogram EBCD = 84 m² + 112 m² = 196 m².                 

For one tile a = 9 cm, b = 28 cm, c = 35 cm
\(\therefore s=\frac { a+b+c }{ 2 } =\frac { 9+28+35 }{ 2 } =36\) cm
\(\therefore \) Area of one tile  \(=\sqrt { s(s-a)(s-b)(s-c) } \)
\(=\sqrt { 36(36-9)(36-28)(36-35) } \)
\(\\ =\sqrt { 36(27)(8)(1) } =\sqrt { 36\left( 9\times 3 \right) \left( 4\times 2 \right) } \)
 \(=6\times 3\times 2\sqrt { 6 } =36\sqrt { 6 } \) cm²
\(\therefore \)  Area of 16 tiles \(=36\sqrt { 6 } \times 16=576\sqrt { 6 } \)cm²
 \(\therefore \) Cost of polishing the tiles at the rate of 50 p per cm².
\(=576\sqrt { 6 } \times 50\) p = Rs. \(\frac { 576\sqrt { 6 } \times 50 }{ 100 } \) 
= Rs. \(288\sqrt { 6 } \) = Rs. 705.60

For one triangular piece = 20 cm, b = 50 cm, c = 50 cm
\(\therefore s=\frac { a+b+c }{ 2 } =\frac { 20+50+50 }{ 2 } =\frac { 120 }{ 2 } =60\) cm
\(\therefore \)  Area of one triangle \(=\sqrt { s(s-a)(s-b)(s-c) } \)
\(=\sqrt { 60(60-20)(60-50)(60-50) } \)
\(=\sqrt { (60)(40)(10)(10) } =200\sqrt { 6 } \) cm²
\(\therefore \)  Area of 5 triangles of one colour \(=5(200\sqrt { 6 } )\) cm² = 1000\(\sqrt { 6 } \) cm²
Hence, 1000\(\sqrt { 6 } \) cm² cloth of each colour is required for the umbrella.

a = 12 cm, b = 12 cm   Perimeter = 30 cm

\(\Rightarrow \)  a +  b + c = 30
\(\Rightarrow \) 12 + 12 + c = 30
 \(\Rightarrow \) 24 + c = 30
 \(\Rightarrow \)  c = 30 - 24
 \(\Rightarrow \) c = 6 cm
\(​​s=\frac { 30 }{ 2 } \) cm = 15 cm
\(\therefore \)  Area of the triangle \(=\sqrt { s(s-a)(s-b)(s-c) } \)
\(=\sqrt { 15(15-12)(15-12)(15-6) } \)
\(=\sqrt { 15(3)(3)(9) } =9\sqrt { 15 } \) cm².

120 cm²

480 cm²

864 cm²

36000 cm²

(a) 114.89 cm²  
(b) \(40\sqrt { 3 } \)cm²  
(c) \(24\sqrt { 3 } \)cm²  
(d) 330 m²

\(100\sqrt { 3 } \)  cm²